diff --git a/core/CHANGELOG.md b/core/CHANGELOG.md
index f9c45f5f..f25ee514 100644
--- a/core/CHANGELOG.md
+++ b/core/CHANGELOG.md
@@ -27,6 +27,8 @@
 
 - Remove `EitherFuture2` in favor of `EitherFuture`. See [PR 3340].
 
+- Remove `EitherOutput` in favor of `future::Either`. See [PR 3341].
+
 [PR 3031]: https://github.com/libp2p/rust-libp2p/pull/3031
 [PR 3058]: https://github.com/libp2p/rust-libp2p/pull/3058
 [PR 3097]: https://github.com/libp2p/rust-libp2p/pull/3097
@@ -36,6 +38,7 @@
 [PR 3338]: https://github.com/libp2p/rust-libp2p/pull/3338
 [PR 3339]: https://github.com/libp2p/rust-libp2p/pull/3339
 [PR 3340]: https://github.com/libp2p/rust-libp2p/pull/3340
+[PR 3341]: https://github.com/libp2p/rust-libp2p/pull/3341
 
 # 0.37.0
 
diff --git a/core/src/either.rs b/core/src/either.rs
index b25e4e12..0641cd78 100644
--- a/core/src/either.rs
+++ b/core/src/either.rs
@@ -25,167 +25,30 @@ use crate::{
     Multiaddr, ProtocolName,
 };
 use either::Either;
-use futures::{
-    io::{IoSlice, IoSliceMut},
-    prelude::*,
-};
+use futures::prelude::*;
 use pin_project::pin_project;
-use std::{io, pin::Pin, task::Context, task::Poll};
+use std::{pin::Pin, task::Context, task::Poll};
 
-/// Implements `AsyncRead` and `AsyncWrite` and dispatches all method calls to
-/// either `First` or `Second`.
-#[pin_project(project = EitherOutputProj)]
-#[derive(Debug, Copy, Clone)]
-pub enum EitherOutput<A, B> {
-    First(#[pin] A),
-    Second(#[pin] B),
-}
-
-impl<A, B> AsyncRead for EitherOutput<A, B>
-where
-    A: AsyncRead,
-    B: AsyncRead,
-{
-    fn poll_read(
-        self: Pin<&mut Self>,
-        cx: &mut Context<'_>,
-        buf: &mut [u8],
-    ) -> Poll<io::Result<usize>> {
-        match self.project() {
-            EitherOutputProj::First(a) => AsyncRead::poll_read(a, cx, buf),
-            EitherOutputProj::Second(b) => AsyncRead::poll_read(b, cx, buf),
-        }
-    }
-
-    fn poll_read_vectored(
-        self: Pin<&mut Self>,
-        cx: &mut Context<'_>,
-        bufs: &mut [IoSliceMut<'_>],
-    ) -> Poll<io::Result<usize>> {
-        match self.project() {
-            EitherOutputProj::First(a) => AsyncRead::poll_read_vectored(a, cx, bufs),
-            EitherOutputProj::Second(b) => AsyncRead::poll_read_vectored(b, cx, bufs),
-        }
-    }
-}
-
-impl<A, B> AsyncWrite for EitherOutput<A, B>
-where
-    A: AsyncWrite,
-    B: AsyncWrite,
-{
-    fn poll_write(
-        self: Pin<&mut Self>,
-        cx: &mut Context<'_>,
-        buf: &[u8],
-    ) -> Poll<io::Result<usize>> {
-        match self.project() {
-            EitherOutputProj::First(a) => AsyncWrite::poll_write(a, cx, buf),
-            EitherOutputProj::Second(b) => AsyncWrite::poll_write(b, cx, buf),
-        }
-    }
-
-    fn poll_write_vectored(
-        self: Pin<&mut Self>,
-        cx: &mut Context<'_>,
-        bufs: &[IoSlice<'_>],
-    ) -> Poll<io::Result<usize>> {
-        match self.project() {
-            EitherOutputProj::First(a) => AsyncWrite::poll_write_vectored(a, cx, bufs),
-            EitherOutputProj::Second(b) => AsyncWrite::poll_write_vectored(b, cx, bufs),
-        }
-    }
-
-    fn poll_flush(self: Pin<&mut Self>, cx: &mut Context<'_>) -> Poll<io::Result<()>> {
-        match self.project() {
-            EitherOutputProj::First(a) => AsyncWrite::poll_flush(a, cx),
-            EitherOutputProj::Second(b) => AsyncWrite::poll_flush(b, cx),
-        }
-    }
-
-    fn poll_close(self: Pin<&mut Self>, cx: &mut Context<'_>) -> Poll<io::Result<()>> {
-        match self.project() {
-            EitherOutputProj::First(a) => AsyncWrite::poll_close(a, cx),
-            EitherOutputProj::Second(b) => AsyncWrite::poll_close(b, cx),
-        }
-    }
-}
-
-impl<A, B, I> Stream for EitherOutput<A, B>
-where
-    A: TryStream<Ok = I>,
-    B: TryStream<Ok = I>,
-{
-    type Item = Result<I, Either<A::Error, B::Error>>;
-
-    fn poll_next(self: Pin<&mut Self>, cx: &mut Context<'_>) -> Poll<Option<Self::Item>> {
-        match self.project() {
-            EitherOutputProj::First(a) => {
-                TryStream::try_poll_next(a, cx).map(|v| v.map(|r| r.map_err(Either::Left)))
-            }
-            EitherOutputProj::Second(b) => {
-                TryStream::try_poll_next(b, cx).map(|v| v.map(|r| r.map_err(Either::Right)))
-            }
-        }
-    }
-}
-
-impl<A, B, I> Sink<I> for EitherOutput<A, B>
-where
-    A: Sink<I>,
-    B: Sink<I>,
-{
-    type Error = Either<A::Error, B::Error>;
-
-    fn poll_ready(self: Pin<&mut Self>, cx: &mut Context<'_>) -> Poll<Result<(), Self::Error>> {
-        match self.project() {
-            EitherOutputProj::First(a) => Sink::poll_ready(a, cx).map_err(Either::Left),
-            EitherOutputProj::Second(b) => Sink::poll_ready(b, cx).map_err(Either::Right),
-        }
-    }
-
-    fn start_send(self: Pin<&mut Self>, item: I) -> Result<(), Self::Error> {
-        match self.project() {
-            EitherOutputProj::First(a) => Sink::start_send(a, item).map_err(Either::Left),
-            EitherOutputProj::Second(b) => Sink::start_send(b, item).map_err(Either::Right),
-        }
-    }
-
-    fn poll_flush(self: Pin<&mut Self>, cx: &mut Context<'_>) -> Poll<Result<(), Self::Error>> {
-        match self.project() {
-            EitherOutputProj::First(a) => Sink::poll_flush(a, cx).map_err(Either::Left),
-            EitherOutputProj::Second(b) => Sink::poll_flush(b, cx).map_err(Either::Right),
-        }
-    }
-
-    fn poll_close(self: Pin<&mut Self>, cx: &mut Context<'_>) -> Poll<Result<(), Self::Error>> {
-        match self.project() {
-            EitherOutputProj::First(a) => Sink::poll_close(a, cx).map_err(Either::Left),
-            EitherOutputProj::Second(b) => Sink::poll_close(b, cx).map_err(Either::Right),
-        }
-    }
-}
-
-impl<A, B> StreamMuxer for EitherOutput<A, B>
+impl<A, B> StreamMuxer for future::Either<A, B>
 where
     A: StreamMuxer,
     B: StreamMuxer,
 {
-    type Substream = EitherOutput<A::Substream, B::Substream>;
+    type Substream = future::Either<A::Substream, B::Substream>;
     type Error = Either<A::Error, B::Error>;
 
     fn poll_inbound(
         self: Pin<&mut Self>,
         cx: &mut Context<'_>,
     ) -> Poll<Result<Self::Substream, Self::Error>> {
-        match self.project() {
-            EitherOutputProj::First(inner) => inner
+        match as_pin_mut(self) {
+            future::Either::Left(inner) => inner
                 .poll_inbound(cx)
-                .map_ok(EitherOutput::First)
+                .map_ok(future::Either::Left)
                 .map_err(Either::Left),
-            EitherOutputProj::Second(inner) => inner
+            future::Either::Right(inner) => inner
                 .poll_inbound(cx)
-                .map_ok(EitherOutput::Second)
+                .map_ok(future::Either::Right)
                 .map_err(Either::Right),
         }
     }
@@ -194,22 +57,22 @@ where
         self: Pin<&mut Self>,
         cx: &mut Context<'_>,
     ) -> Poll<Result<Self::Substream, Self::Error>> {
-        match self.project() {
-            EitherOutputProj::First(inner) => inner
+        match as_pin_mut(self) {
+            future::Either::Left(inner) => inner
                 .poll_outbound(cx)
-                .map_ok(EitherOutput::First)
+                .map_ok(future::Either::Left)
                 .map_err(Either::Left),
-            EitherOutputProj::Second(inner) => inner
+            future::Either::Right(inner) => inner
                 .poll_outbound(cx)
-                .map_ok(EitherOutput::Second)
+                .map_ok(future::Either::Right)
                 .map_err(Either::Right),
         }
     }
 
     fn poll_close(self: Pin<&mut Self>, cx: &mut Context<'_>) -> Poll<Result<(), Self::Error>> {
-        match self.project() {
-            EitherOutputProj::First(inner) => inner.poll_close(cx).map_err(Either::Left),
-            EitherOutputProj::Second(inner) => inner.poll_close(cx).map_err(Either::Right),
+        match as_pin_mut(self) {
+            future::Either::Left(inner) => inner.poll_close(cx).map_err(Either::Left),
+            future::Either::Right(inner) => inner.poll_close(cx).map_err(Either::Right),
         }
     }
 
@@ -217,9 +80,31 @@ where
         self: Pin<&mut Self>,
         cx: &mut Context<'_>,
     ) -> Poll<Result<StreamMuxerEvent, Self::Error>> {
-        match self.project() {
-            EitherOutputProj::First(inner) => inner.poll(cx).map_err(Either::Left),
-            EitherOutputProj::Second(inner) => inner.poll(cx).map_err(Either::Right),
+        match as_pin_mut(self) {
+            future::Either::Left(inner) => inner.poll(cx).map_err(Either::Left),
+            future::Either::Right(inner) => inner.poll(cx).map_err(Either::Right),
+        }
+    }
+}
+
+/// Convert `Pin<&mut Either<A, B>>` to `Either<Pin<&mut A>, Pin<&mut B>>`,
+/// pinned projections of the inner variants.
+///
+/// Local function until <https://github.com/rust-lang/futures-rs/pull/2691> is merged.
+fn as_pin_mut<A, B>(
+    either: Pin<&mut future::Either<A, B>>,
+) -> future::Either<Pin<&mut A>, Pin<&mut B>> {
+    // SAFETY: `get_unchecked_mut` is fine because we don't move anything.
+    // We can use `new_unchecked` because the `inner` parts are guaranteed
+    // to be pinned, as they come from `self` which is pinned, and we never
+    // offer an unpinned `&mut L` or `&mut R` through `Pin<&mut Self>`. We
+    // also don't have an implementation of `Drop`, nor manual `Unpin`.
+    unsafe {
+        match *Pin::get_unchecked_mut(either) {
+            future::Either::Left(ref mut inner) => future::Either::Left(Pin::new_unchecked(inner)),
+            future::Either::Right(ref mut inner) => {
+                future::Either::Right(Pin::new_unchecked(inner))
+            }
         }
     }
 }
@@ -238,15 +123,15 @@ where
     AFuture: TryFuture<Ok = AInner>,
     BFuture: TryFuture<Ok = BInner>,
 {
-    type Output = Result<EitherOutput<AInner, BInner>, Either<AFuture::Error, BFuture::Error>>;
+    type Output = Result<future::Either<AInner, BInner>, Either<AFuture::Error, BFuture::Error>>;
 
     fn poll(self: Pin<&mut Self>, cx: &mut Context<'_>) -> Poll<Self::Output> {
         match self.project() {
             EitherFutureProj::First(a) => TryFuture::try_poll(a, cx)
-                .map_ok(EitherOutput::First)
+                .map_ok(future::Either::Left)
                 .map_err(Either::Left),
             EitherFutureProj::Second(a) => TryFuture::try_poll(a, cx)
-                .map_ok(EitherOutput::Second)
+                .map_ok(future::Either::Right)
                 .map_err(Either::Right),
         }
     }
@@ -272,7 +157,7 @@ where
     B: Transport,
     A: Transport,
 {
-    type Output = EitherOutput<A::Output, B::Output>;
+    type Output = future::Either<A::Output, B::Output>;
     type Error = Either<A::Error, B::Error>;
     type ListenerUpgrade = EitherFuture<A::ListenerUpgrade, B::ListenerUpgrade>;
     type Dial = EitherFuture<A::Dial, B::Dial>;
diff --git a/core/src/transport/choice.rs b/core/src/transport/choice.rs
index 2405f013..bb7d542d 100644
--- a/core/src/transport/choice.rs
+++ b/core/src/transport/choice.rs
@@ -18,9 +18,10 @@
 // FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
 // DEALINGS IN THE SOFTWARE.
 
-use crate::either::{EitherFuture, EitherOutput};
+use crate::either::EitherFuture;
 use crate::transport::{ListenerId, Transport, TransportError, TransportEvent};
 use either::Either;
+use futures::future;
 use multiaddr::Multiaddr;
 use std::{pin::Pin, task::Context, task::Poll};
 
@@ -40,7 +41,7 @@ where
     B: Transport,
     A: Transport,
 {
-    type Output = EitherOutput<A::Output, B::Output>;
+    type Output = future::Either<A::Output, B::Output>;
     type Error = Either<A::Error, B::Error>;
     type ListenerUpgrade = EitherFuture<A::ListenerUpgrade, B::ListenerUpgrade>;
     type Dial = EitherFuture<A::Dial, B::Dial>;
diff --git a/core/src/upgrade/either.rs b/core/src/upgrade/either.rs
index 5785fda2..563e7651 100644
--- a/core/src/upgrade/either.rs
+++ b/core/src/upgrade/either.rs
@@ -19,10 +19,11 @@
 // DEALINGS IN THE SOFTWARE.
 
 use crate::{
-    either::{EitherFuture, EitherName, EitherOutput},
+    either::{EitherFuture, EitherName},
     upgrade::{InboundUpgrade, OutboundUpgrade, UpgradeInfo},
 };
 use either::Either;
+use futures::future;
 
 impl<A, B> UpgradeInfo for Either<A, B>
 where
@@ -48,7 +49,7 @@ where
     A: InboundUpgrade<C, Output = TA, Error = EA>,
     B: InboundUpgrade<C, Output = TB, Error = EB>,
 {
-    type Output = EitherOutput<TA, TB>;
+    type Output = future::Either<TA, TB>;
     type Error = Either<EA, EB>;
     type Future = EitherFuture<A::Future, B::Future>;
 
@@ -70,7 +71,7 @@ where
     A: OutboundUpgrade<C, Output = TA, Error = EA>,
     B: OutboundUpgrade<C, Output = TB, Error = EB>,
 {
-    type Output = EitherOutput<TA, TB>;
+    type Output = future::Either<TA, TB>;
     type Error = Either<EA, EB>;
     type Future = EitherFuture<A::Future, B::Future>;
 
diff --git a/core/src/upgrade/select.rs b/core/src/upgrade/select.rs
index cb3cdc23..1d5ab9ab 100644
--- a/core/src/upgrade/select.rs
+++ b/core/src/upgrade/select.rs
@@ -20,10 +20,11 @@
 
 use crate::either::EitherFuture;
 use crate::{
-    either::{EitherName, EitherOutput},
+    either::EitherName,
     upgrade::{InboundUpgrade, OutboundUpgrade, UpgradeInfo},
 };
 use either::Either;
+use futures::future;
 
 /// Upgrade that combines two upgrades into one. Supports all the protocols supported by either
 /// sub-upgrade.
@@ -65,7 +66,7 @@ where
     A: InboundUpgrade<C, Output = TA, Error = EA>,
     B: InboundUpgrade<C, Output = TB, Error = EB>,
 {
-    type Output = EitherOutput<TA, TB>;
+    type Output = future::Either<TA, TB>;
     type Error = Either<EA, EB>;
     type Future = EitherFuture<A::Future, B::Future>;
 
@@ -82,7 +83,7 @@ where
     A: OutboundUpgrade<C, Output = TA, Error = EA>,
     B: OutboundUpgrade<C, Output = TB, Error = EB>,
 {
-    type Output = EitherOutput<TA, TB>;
+    type Output = future::Either<TA, TB>;
     type Error = Either<EA, EB>;
     type Future = EitherFuture<A::Future, B::Future>;
 
diff --git a/protocols/dcutr/src/handler/relayed.rs b/protocols/dcutr/src/handler/relayed.rs
index 23a45841..0b487c30 100644
--- a/protocols/dcutr/src/handler/relayed.rs
+++ b/protocols/dcutr/src/handler/relayed.rs
@@ -22,9 +22,9 @@
 
 use crate::protocol;
 use either::Either;
+use futures::future;
 use futures::future::{BoxFuture, FutureExt};
 use instant::Instant;
-use libp2p_core::either::EitherOutput;
 use libp2p_core::multiaddr::Multiaddr;
 use libp2p_core::upgrade::{DeniedUpgrade, NegotiationError, UpgradeError};
 use libp2p_core::ConnectedPoint;
@@ -174,7 +174,7 @@ impl Handler {
         >,
     ) {
         match output {
-            EitherOutput::First(inbound_connect) => {
+            future::Either::Left(inbound_connect) => {
                 let remote_addr = match &self.endpoint {
                     ConnectedPoint::Dialer { address, role_override: _ } => address.clone(),
                     ConnectedPoint::Listener { ..} => unreachable!("`<Handler as ConnectionHandler>::listen_protocol` denies all incoming substreams as a listener."),
@@ -187,7 +187,7 @@ impl Handler {
                 ));
             }
             // A connection listener denies all incoming substreams, thus none can ever be fully negotiated.
-            EitherOutput::Second(output) => void::unreachable(output),
+            future::Either::Right(output) => void::unreachable(output),
         }
     }
 
diff --git a/protocols/identify/src/handler.rs b/protocols/identify/src/handler.rs
index e251d552..7b5f4b58 100644
--- a/protocols/identify/src/handler.rs
+++ b/protocols/identify/src/handler.rs
@@ -26,7 +26,6 @@ use futures::future::BoxFuture;
 use futures::prelude::*;
 use futures::stream::FuturesUnordered;
 use futures_timer::Delay;
-use libp2p_core::either::EitherOutput;
 use libp2p_core::upgrade::SelectUpgrade;
 use libp2p_core::{ConnectedPoint, Multiaddr, PeerId, PublicKey};
 use libp2p_swarm::handler::{
@@ -201,7 +200,7 @@ impl Handler {
         >,
     ) {
         match output {
-            EitherOutput::First(substream) => {
+            future::Either::Left(substream) => {
                 self.events
                     .push(ConnectionHandlerEvent::Custom(Event::Identify));
                 if !self.reply_streams.is_empty() {
@@ -213,7 +212,7 @@ impl Handler {
                 }
                 self.reply_streams.push_back(substream);
             }
-            EitherOutput::Second(fut) => {
+            future::Either::Right(fut) => {
                 if self.inbound_identify_push.replace(fut).is_some() {
                     warn!(
                         "New inbound identify push stream from {} while still \
@@ -235,14 +234,14 @@ impl Handler {
         >,
     ) {
         match output {
-            EitherOutput::First(remote_info) => {
+            future::Either::Left(remote_info) => {
                 self.events
                     .push(ConnectionHandlerEvent::Custom(Event::Identified(
                         remote_info,
                     )));
                 self.keep_alive = KeepAlive::No;
             }
-            EitherOutput::Second(()) => self
+            future::Either::Right(()) => self
                 .events
                 .push(ConnectionHandlerEvent::Custom(Event::IdentificationPushed)),
         }
diff --git a/protocols/kad/src/handler.rs b/protocols/kad/src/handler.rs
index a475c045..1c051624 100644
--- a/protocols/kad/src/handler.rs
+++ b/protocols/kad/src/handler.rs
@@ -27,7 +27,7 @@ use either::Either;
 use futures::prelude::*;
 use futures::stream::SelectAll;
 use instant::Instant;
-use libp2p_core::{either::EitherOutput, upgrade, ConnectedPoint, PeerId};
+use libp2p_core::{upgrade, ConnectedPoint, PeerId};
 use libp2p_swarm::handler::{
     ConnectionEvent, DialUpgradeError, FullyNegotiatedInbound, FullyNegotiatedOutbound,
 };
@@ -561,8 +561,8 @@ where
         // If `self.allow_listening` is false, then we produced a `DeniedUpgrade` and `protocol`
         // is a `Void`.
         let protocol = match protocol {
-            EitherOutput::First(p) => p,
-            EitherOutput::Second(p) => void::unreachable(p),
+            future::Either::Left(p) => p,
+            future::Either::Right(p) => void::unreachable(p),
         };
 
         if let ProtocolStatus::Unconfirmed = self.protocol_status {
diff --git a/swarm-derive/src/lib.rs b/swarm-derive/src/lib.rs
index 489ec8c7..a95d99a3 100644
--- a/swarm-derive/src/lib.rs
+++ b/swarm-derive/src/lib.rs
@@ -53,7 +53,7 @@ fn build_struct(ast: &DeriveInput, data_struct: &DataStruct) -> TokenStream {
 
     let multiaddr = quote! { #prelude_path::Multiaddr };
     let trait_to_impl = quote! { #prelude_path::NetworkBehaviour };
-    let either_ident = quote! { #prelude_path::EitherOutput };
+    let either_ident = quote! { #prelude_path::Either };
     let network_behaviour_action = quote! { #prelude_path::NetworkBehaviourAction };
     let into_connection_handler = quote! { #prelude_path::IntoConnectionHandler };
     let connection_handler = quote! { #prelude_path::ConnectionHandler };
@@ -531,13 +531,13 @@ fn build_struct(ast: &DeriveInput, data_struct: &DataStruct) -> TokenStream {
             .enumerate()
             .map(|(enum_n, (field_n, field))| {
                 let mut elem = if enum_n != 0 {
-                    quote! { #either_ident::Second(ev) }
+                    quote! { #either_ident::Right(ev) }
                 } else {
                     quote! { ev }
                 };
 
                 for _ in 0..data_struct.fields.len() - 1 - enum_n {
-                    elem = quote! { #either_ident::First(#elem) };
+                    elem = quote! { #either_ident::Left(#elem) };
                 }
 
                 Some(match field.ident {
@@ -599,12 +599,12 @@ fn build_struct(ast: &DeriveInput, data_struct: &DataStruct) -> TokenStream {
             .expect("Fields of NetworkBehaviour implementation to be named.");
 
         let mut wrapped_event = if field_n != 0 {
-            quote!{ #either_ident::Second(event) }
+            quote!{ #either_ident::Right(event) }
         } else {
             quote!{ event }
         };
         for _ in 0 .. data_struct.fields.len() - 1 - field_n {
-            wrapped_event = quote!{ #either_ident::First(#wrapped_event) };
+            wrapped_event = quote!{ #either_ident::Left(#wrapped_event) };
         }
 
         // `Dial` provides a handler of the specific behaviour triggering the
diff --git a/swarm/src/behaviour/toggle.rs b/swarm/src/behaviour/toggle.rs
index 897d515c..8a384127 100644
--- a/swarm/src/behaviour/toggle.rs
+++ b/swarm/src/behaviour/toggle.rs
@@ -28,9 +28,8 @@ use crate::handler::{
 use crate::upgrade::SendWrapper;
 use crate::{NetworkBehaviour, NetworkBehaviourAction, PollParameters};
 use either::Either;
-use libp2p_core::{
-    either::EitherOutput, upgrade::DeniedUpgrade, ConnectedPoint, Multiaddr, PeerId,
-};
+use futures::future;
+use libp2p_core::{upgrade::DeniedUpgrade, ConnectedPoint, Multiaddr, PeerId};
 use std::{task::Context, task::Poll};
 
 /// Implementation of `NetworkBehaviour` that can be either in the disabled or enabled state.
@@ -169,8 +168,8 @@ where
         >,
     ) {
         let out = match out {
-            EitherOutput::First(out) => out,
-            EitherOutput::Second(v) => void::unreachable(v),
+            future::Either::Left(out) => out,
+            future::Either::Right(v) => void::unreachable(v),
         };
 
         if let Either::Left(info) = info {
diff --git a/swarm/src/handler/either.rs b/swarm/src/handler/either.rs
index 2345830e..e2c72bb3 100644
--- a/swarm/src/handler/either.rs
+++ b/swarm/src/handler/either.rs
@@ -20,12 +20,12 @@
 
 use crate::handler::{
     ConnectionEvent, ConnectionHandler, ConnectionHandlerEvent, ConnectionHandlerUpgrErr,
-    DialUpgradeError, FullyNegotiatedInbound, FullyNegotiatedOutbound, InboundUpgradeSend,
-    IntoConnectionHandler, KeepAlive, ListenUpgradeError, OutboundUpgradeSend, SubstreamProtocol,
+    FullyNegotiatedInbound, InboundUpgradeSend, IntoConnectionHandler, KeepAlive,
+    ListenUpgradeError, SubstreamProtocol,
 };
 use crate::upgrade::SendWrapper;
 use either::Either;
-use libp2p_core::either::EitherOutput;
+use futures::future;
 use libp2p_core::upgrade::UpgradeError;
 use libp2p_core::{ConnectedPoint, PeerId};
 use std::task::{Context, Poll};
@@ -101,11 +101,11 @@ where
     ) -> Either<FullyNegotiatedInbound<LIP, LIOI>, FullyNegotiatedInbound<RIP, RIOI>> {
         match self {
             FullyNegotiatedInbound {
-                protocol: EitherOutput::First(protocol),
+                protocol: future::Either::Left(protocol),
                 info: Either::Left(info),
             } => Either::Left(FullyNegotiatedInbound { protocol, info }),
             FullyNegotiatedInbound {
-                protocol: EitherOutput::Second(protocol),
+                protocol: future::Either::Right(protocol),
                 info: Either::Right(info),
             } => Either::Right(FullyNegotiatedInbound { protocol, info }),
             _ => unreachable!(),
@@ -113,98 +113,6 @@ where
     }
 }
 
-impl<LOP, ROP, LOOI, ROOI>
-    FullyNegotiatedOutbound<Either<SendWrapper<LOP>, SendWrapper<ROP>>, Either<LOOI, ROOI>>
-where
-    LOP: OutboundUpgradeSend,
-    ROP: OutboundUpgradeSend,
-{
-    fn transpose(
-        self,
-    ) -> Either<FullyNegotiatedOutbound<LOP, LOOI>, FullyNegotiatedOutbound<ROP, ROOI>> {
-        match self {
-            FullyNegotiatedOutbound {
-                protocol: EitherOutput::First(protocol),
-                info: Either::Left(info),
-            } => Either::Left(FullyNegotiatedOutbound { protocol, info }),
-            FullyNegotiatedOutbound {
-                protocol: EitherOutput::Second(protocol),
-                info: Either::Right(info),
-            } => Either::Right(FullyNegotiatedOutbound { protocol, info }),
-            _ => unreachable!(),
-        }
-    }
-}
-
-impl<LOP, ROP, LOOI, ROOI>
-    DialUpgradeError<Either<LOOI, ROOI>, Either<SendWrapper<LOP>, SendWrapper<ROP>>>
-where
-    LOP: OutboundUpgradeSend,
-    ROP: OutboundUpgradeSend,
-{
-    fn transpose(self) -> Either<DialUpgradeError<LOOI, LOP>, DialUpgradeError<ROOI, ROP>> {
-        match self {
-            DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Upgrade(UpgradeError::Apply(Either::Left(error))),
-                info: Either::Left(info),
-            } => Either::Left(DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Upgrade(UpgradeError::Apply(error)),
-                info,
-            }),
-            DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Upgrade(UpgradeError::Apply(Either::Right(error))),
-                info: Either::Right(info),
-            } => Either::Right(DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Upgrade(UpgradeError::Apply(error)),
-                info,
-            }),
-            DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Upgrade(UpgradeError::Select(error)),
-                info: Either::Left(info),
-            } => Either::Left(DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Upgrade(UpgradeError::Select(error)),
-                info,
-            }),
-            DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Upgrade(UpgradeError::Select(error)),
-                info: Either::Right(info),
-            } => Either::Right(DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Upgrade(UpgradeError::Select(error)),
-                info,
-            }),
-            DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Timer,
-                info: Either::Left(info),
-            } => Either::Left(DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Timer,
-                info,
-            }),
-            DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Timer,
-                info: Either::Right(info),
-            } => Either::Right(DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Timer,
-                info,
-            }),
-            DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Timeout,
-                info: Either::Left(info),
-            } => Either::Left(DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Timeout,
-                info,
-            }),
-            DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Timeout,
-                info: Either::Right(info),
-            } => Either::Right(DialUpgradeError {
-                error: ConnectionHandlerUpgrErr::Timeout,
-                info,
-            }),
-            _ => unreachable!(),
-        }
-    }
-}
-
 impl<LIP, RIP, LIOI, RIOI>
     ListenUpgradeError<Either<LIOI, RIOI>, Either<SendWrapper<LIP>, SendWrapper<RIP>>>
 where
diff --git a/swarm/src/handler/select.rs b/swarm/src/handler/select.rs
index 535b06be..8c38ffe7 100644
--- a/swarm/src/handler/select.rs
+++ b/swarm/src/handler/select.rs
@@ -27,8 +27,8 @@ use crate::handler::{
 use crate::upgrade::SendWrapper;
 
 use either::Either;
+use futures::future;
 use libp2p_core::{
-    either::EitherOutput,
     upgrade::{NegotiationError, ProtocolError, SelectUpgrade, UpgradeError},
     ConnectedPoint, PeerId,
 };
@@ -101,27 +101,24 @@ impl<TProto1, TProto2> ConnectionHandlerSelect<TProto1, TProto2> {
 }
 
 impl<S1OOI, S2OOI, S1OP, S2OP>
-    FullyNegotiatedOutbound<
-        Either<SendWrapper<S1OP>, SendWrapper<S2OP>>,
-        EitherOutput<S1OOI, S2OOI>,
-    >
+    FullyNegotiatedOutbound<Either<SendWrapper<S1OP>, SendWrapper<S2OP>>, Either<S1OOI, S2OOI>>
 where
     S1OP: OutboundUpgradeSend,
     S2OP: OutboundUpgradeSend,
     S1OOI: Send + 'static,
     S2OOI: Send + 'static,
 {
-    fn transpose(
+    pub(crate) fn transpose(
         self,
     ) -> Either<FullyNegotiatedOutbound<S1OP, S1OOI>, FullyNegotiatedOutbound<S2OP, S2OOI>> {
         match self {
             FullyNegotiatedOutbound {
-                protocol: EitherOutput::First(protocol),
-                info: EitherOutput::First(info),
+                protocol: future::Either::Left(protocol),
+                info: Either::Left(info),
             } => Either::Left(FullyNegotiatedOutbound { protocol, info }),
             FullyNegotiatedOutbound {
-                protocol: EitherOutput::Second(protocol),
-                info: EitherOutput::Second(info),
+                protocol: future::Either::Right(protocol),
+                info: Either::Right(info),
             } => Either::Right(FullyNegotiatedOutbound { protocol, info }),
             _ => panic!("wrong API usage: the protocol doesn't match the upgrade info"),
         }
@@ -134,16 +131,16 @@ where
     S1IP: InboundUpgradeSend,
     S2IP: InboundUpgradeSend,
 {
-    fn transpose(
+    pub(crate) fn transpose(
         self,
     ) -> Either<FullyNegotiatedInbound<S1IP, S1IOI>, FullyNegotiatedInbound<S2IP, S2IOI>> {
         match self {
             FullyNegotiatedInbound {
-                protocol: EitherOutput::First(protocol),
+                protocol: future::Either::Left(protocol),
                 info: (i1, _i2),
             } => Either::Left(FullyNegotiatedInbound { protocol, info: i1 }),
             FullyNegotiatedInbound {
-                protocol: EitherOutput::Second(protocol),
+                protocol: future::Either::Right(protocol),
                 info: (_i1, i2),
             } => Either::Right(FullyNegotiatedInbound { protocol, info: i2 }),
         }
@@ -151,66 +148,68 @@ where
 }
 
 impl<S1OOI, S2OOI, S1OP, S2OP>
-    DialUpgradeError<EitherOutput<S1OOI, S2OOI>, Either<SendWrapper<S1OP>, SendWrapper<S2OP>>>
+    DialUpgradeError<Either<S1OOI, S2OOI>, Either<SendWrapper<S1OP>, SendWrapper<S2OP>>>
 where
     S1OP: OutboundUpgradeSend,
     S2OP: OutboundUpgradeSend,
     S1OOI: Send + 'static,
     S2OOI: Send + 'static,
 {
-    fn transpose(self) -> Either<DialUpgradeError<S1OOI, S1OP>, DialUpgradeError<S2OOI, S2OP>> {
+    pub(crate) fn transpose(
+        self,
+    ) -> Either<DialUpgradeError<S1OOI, S1OP>, DialUpgradeError<S2OOI, S2OP>> {
         match self {
             DialUpgradeError {
-                info: EitherOutput::First(info),
+                info: Either::Left(info),
                 error: ConnectionHandlerUpgrErr::Timer,
             } => Either::Left(DialUpgradeError {
                 info,
                 error: ConnectionHandlerUpgrErr::Timer,
             }),
             DialUpgradeError {
-                info: EitherOutput::First(info),
+                info: Either::Left(info),
                 error: ConnectionHandlerUpgrErr::Timeout,
             } => Either::Left(DialUpgradeError {
                 info,
                 error: ConnectionHandlerUpgrErr::Timeout,
             }),
             DialUpgradeError {
-                info: EitherOutput::First(info),
+                info: Either::Left(info),
                 error: ConnectionHandlerUpgrErr::Upgrade(UpgradeError::Select(err)),
             } => Either::Left(DialUpgradeError {
                 info,
                 error: ConnectionHandlerUpgrErr::Upgrade(UpgradeError::Select(err)),
             }),
             DialUpgradeError {
-                info: EitherOutput::First(info),
+                info: Either::Left(info),
                 error: ConnectionHandlerUpgrErr::Upgrade(UpgradeError::Apply(Either::Left(err))),
             } => Either::Left(DialUpgradeError {
                 info,
                 error: ConnectionHandlerUpgrErr::Upgrade(UpgradeError::Apply(err)),
             }),
             DialUpgradeError {
-                info: EitherOutput::Second(info),
+                info: Either::Right(info),
                 error: ConnectionHandlerUpgrErr::Timer,
             } => Either::Right(DialUpgradeError {
                 info,
                 error: ConnectionHandlerUpgrErr::Timer,
             }),
             DialUpgradeError {
-                info: EitherOutput::Second(info),
+                info: Either::Right(info),
                 error: ConnectionHandlerUpgrErr::Timeout,
             } => Either::Right(DialUpgradeError {
                 info,
                 error: ConnectionHandlerUpgrErr::Timeout,
             }),
             DialUpgradeError {
-                info: EitherOutput::Second(info),
+                info: Either::Right(info),
                 error: ConnectionHandlerUpgrErr::Upgrade(UpgradeError::Select(err)),
             } => Either::Right(DialUpgradeError {
                 info,
                 error: ConnectionHandlerUpgrErr::Upgrade(UpgradeError::Select(err)),
             }),
             DialUpgradeError {
-                info: EitherOutput::Second(info),
+                info: Either::Right(info),
                 error: ConnectionHandlerUpgrErr::Upgrade(UpgradeError::Apply(Either::Right(err))),
             } => Either::Right(DialUpgradeError {
                 info,
@@ -338,8 +337,8 @@ where
     TProto1: ConnectionHandler,
     TProto2: ConnectionHandler,
 {
-    type InEvent = EitherOutput<TProto1::InEvent, TProto2::InEvent>;
-    type OutEvent = EitherOutput<TProto1::OutEvent, TProto2::OutEvent>;
+    type InEvent = Either<TProto1::InEvent, TProto2::InEvent>;
+    type OutEvent = Either<TProto1::OutEvent, TProto2::OutEvent>;
     type Error = Either<TProto1::Error, TProto2::Error>;
     type InboundProtocol = SelectUpgrade<
         SendWrapper<<TProto1 as ConnectionHandler>::InboundProtocol>,
@@ -347,7 +346,7 @@ where
     >;
     type OutboundProtocol =
         Either<SendWrapper<TProto1::OutboundProtocol>, SendWrapper<TProto2::OutboundProtocol>>;
-    type OutboundOpenInfo = EitherOutput<TProto1::OutboundOpenInfo, TProto2::OutboundOpenInfo>;
+    type OutboundOpenInfo = Either<TProto1::OutboundOpenInfo, TProto2::OutboundOpenInfo>;
     type InboundOpenInfo = (TProto1::InboundOpenInfo, TProto2::InboundOpenInfo);
 
     fn listen_protocol(&self) -> SubstreamProtocol<Self::InboundProtocol, Self::InboundOpenInfo> {
@@ -362,8 +361,8 @@ where
 
     fn on_behaviour_event(&mut self, event: Self::InEvent) {
         match event {
-            EitherOutput::First(event) => self.proto1.on_behaviour_event(event),
-            EitherOutput::Second(event) => self.proto2.on_behaviour_event(event),
+            Either::Left(event) => self.proto1.on_behaviour_event(event),
+            Either::Right(event) => self.proto2.on_behaviour_event(event),
         }
     }
 
@@ -387,7 +386,7 @@ where
     > {
         match self.proto1.poll(cx) {
             Poll::Ready(ConnectionHandlerEvent::Custom(event)) => {
-                return Poll::Ready(ConnectionHandlerEvent::Custom(EitherOutput::First(event)));
+                return Poll::Ready(ConnectionHandlerEvent::Custom(Either::Left(event)));
             }
             Poll::Ready(ConnectionHandlerEvent::Close(event)) => {
                 return Poll::Ready(ConnectionHandlerEvent::Close(Either::Left(event)));
@@ -396,7 +395,7 @@ where
                 return Poll::Ready(ConnectionHandlerEvent::OutboundSubstreamRequest {
                     protocol: protocol
                         .map_upgrade(|u| Either::Left(SendWrapper(u)))
-                        .map_info(EitherOutput::First),
+                        .map_info(Either::Left),
                 });
             }
             Poll::Pending => (),
@@ -404,7 +403,7 @@ where
 
         match self.proto2.poll(cx) {
             Poll::Ready(ConnectionHandlerEvent::Custom(event)) => {
-                return Poll::Ready(ConnectionHandlerEvent::Custom(EitherOutput::Second(event)));
+                return Poll::Ready(ConnectionHandlerEvent::Custom(Either::Right(event)));
             }
             Poll::Ready(ConnectionHandlerEvent::Close(event)) => {
                 return Poll::Ready(ConnectionHandlerEvent::Close(Either::Right(event)));
@@ -413,7 +412,7 @@ where
                 return Poll::Ready(ConnectionHandlerEvent::OutboundSubstreamRequest {
                     protocol: protocol
                         .map_upgrade(|u| Either::Right(SendWrapper(u)))
-                        .map_info(EitherOutput::Second),
+                        .map_info(Either::Right),
                 });
             }
             Poll::Pending => (),
diff --git a/swarm/src/lib.rs b/swarm/src/lib.rs
index 0e2b0abc..ee8a2808 100644
--- a/swarm/src/lib.rs
+++ b/swarm/src/lib.rs
@@ -92,8 +92,8 @@ pub mod derive_prelude {
     pub use crate::NetworkBehaviour;
     pub use crate::NetworkBehaviourAction;
     pub use crate::PollParameters;
+    pub use either::Either;
     pub use futures::prelude as futures;
-    pub use libp2p_core::either::EitherOutput;
     pub use libp2p_core::transport::ListenerId;
     pub use libp2p_core::ConnectedPoint;
     pub use libp2p_core::Multiaddr;
diff --git a/transports/quic/tests/smoke.rs b/transports/quic/tests/smoke.rs
index a1478645..7950bbdc 100644
--- a/transports/quic/tests/smoke.rs
+++ b/transports/quic/tests/smoke.rs
@@ -4,7 +4,6 @@ use futures::channel::{mpsc, oneshot};
 use futures::future::{poll_fn, Either};
 use futures::stream::StreamExt;
 use futures::{future, AsyncReadExt, AsyncWriteExt, FutureExt, SinkExt};
-use libp2p_core::either::EitherOutput;
 use libp2p_core::muxing::{StreamMuxerBox, StreamMuxerExt, SubstreamBox};
 use libp2p_core::transport::{Boxed, OrTransport, TransportEvent};
 use libp2p_core::{multiaddr::Protocol, upgrade, Multiaddr, PeerId, Transport};
@@ -131,8 +130,8 @@ fn new_tcp_quic_transport() -> (PeerId, Boxed<(PeerId, StreamMuxerBox)>) {
 
     let transport = OrTransport::new(quic_transport, tcp_transport)
         .map(|either_output, _| match either_output {
-            EitherOutput::First((peer_id, muxer)) => (peer_id, StreamMuxerBox::new(muxer)),
-            EitherOutput::Second((peer_id, muxer)) => (peer_id, StreamMuxerBox::new(muxer)),
+            Either::Left((peer_id, muxer)) => (peer_id, StreamMuxerBox::new(muxer)),
+            Either::Right((peer_id, muxer)) => (peer_id, StreamMuxerBox::new(muxer)),
         })
         .boxed();
 
diff --git a/transports/websocket/src/framed.rs b/transports/websocket/src/framed.rs
index 0b069d27..318090d1 100644
--- a/transports/websocket/src/framed.rs
+++ b/transports/websocket/src/framed.rs
@@ -24,7 +24,6 @@ use futures::{future::BoxFuture, prelude::*, ready, stream::BoxStream};
 use futures_rustls::{client, rustls, server};
 use libp2p_core::{
     connection::Endpoint,
-    either::EitherOutput,
     multiaddr::{Multiaddr, Protocol},
     transport::{ListenerId, TransportError, TransportEvent},
     Transport,
@@ -108,7 +107,7 @@ impl<T> WsConfig<T> {
     }
 }
 
-type TlsOrPlain<T> = EitherOutput<EitherOutput<client::TlsStream<T>, server::TlsStream<T>>, T>;
+type TlsOrPlain<T> = future::Either<future::Either<client::TlsStream<T>, server::TlsStream<T>>, T>;
 
 impl<T> Transport for WsConfig<T>
 where
@@ -350,11 +349,11 @@ where
                 })
                 .await?;
 
-            let stream: TlsOrPlain<_> = EitherOutput::First(EitherOutput::First(stream));
+            let stream: TlsOrPlain<_> = future::Either::Left(future::Either::Left(stream));
             stream
         } else {
             // continue with plain stream
-            EitherOutput::Second(stream)
+            future::Either::Right(stream)
         };
 
         trace!("Sending websocket handshake to {}", addr.host_port);
@@ -422,12 +421,12 @@ where
                     })
                     .await?;
 
-                let stream: TlsOrPlain<_> = EitherOutput::First(EitherOutput::Second(stream));
+                let stream: TlsOrPlain<_> = future::Either::Left(future::Either::Right(stream));
 
                 stream
             } else {
                 // continue with plain stream
-                EitherOutput::Second(stream)
+                future::Either::Right(stream)
             };
 
             trace!(