diff --git a/client_test.go b/client_test.go
index 79bc3a0..66c0a51 100644
--- a/client_test.go
+++ b/client_test.go
@@ -109,3 +109,33 @@ func testExecerFail(ctx context.Context, t *testing.T, execer Execer) {
 	assert.True(t, "exit code is nonzero", code.Code != 0)
 	assert.Error(t, "wait for process to error", err)
 }
+
+func TestStderrVsStdout(t *testing.T) {
+	ctx, cancel := context.WithTimeout(context.Background(), 15*time.Second)
+	defer cancel()
+
+	var (
+		stdout bytes.Buffer
+		stderr bytes.Buffer
+	)
+
+	ws, server := mockConn(ctx, t)
+	defer server.Close()
+
+	execer := RemoteExecer(ws)
+	process, err := execer.Start(ctx, Command{
+		Command: "sh",
+		Args:    []string{"-c", "echo stdout-message; echo 1>&2 stderr-message"},
+		Stdin:   false,
+	})
+	assert.Success(t, "start command", err)
+
+	go io.Copy(&stdout, process.Stdout())
+	go io.Copy(&stderr, process.Stderr())
+
+	err = process.Wait()
+	assert.Success(t, "wait for process to complete", err)
+
+	assert.Equal(t, "stdout", "stdout-message", strings.TrimSpace(stdout.String()))
+	assert.Equal(t, "stderr", "stderr-message", strings.TrimSpace(stderr.String()))
+}
diff --git a/localexec_test.go b/localexec_test.go
index d2e6564..b33cb27 100644
--- a/localexec_test.go
+++ b/localexec_test.go
@@ -130,20 +130,18 @@ func TestStdoutVsStderr(t *testing.T) {
 	)
 	process, err := execer.Start(ctx, Command{
 		Command: "sh",
-		Args: []string{"-c", "echo stdout-message; echo 1>&2 stderr-message"},
-		Stdin: false,
-		TTY: false,
+		Args:    []string{"-c", "echo stdout-message; echo 1>&2 stderr-message"},
+		Stdin:   false,
+		TTY:     false,
 	})
 	assert.Success(t, "start command", err)
 
 	go io.Copy(&stdout, process.Stdout())
 	go io.Copy(&stderr, process.Stderr())
 
-	time.Sleep(time.Second)
 	err = process.Wait()
 	assert.Success(t, "wait for process to complete", err)
 
 	assert.Equal(t, "stdout", "stdout-message", strings.TrimSpace(stdout.String()))
 	assert.Equal(t, "stderr", "stderr-message", strings.TrimSpace(stderr.String()))
 }
-
diff --git a/server.go b/server.go
index 05a9518..224a6c6 100644
--- a/server.go
+++ b/server.go
@@ -78,7 +78,7 @@ func Serve(ctx context.Context, c *websocket.Conn, execer Execer) error {
 				return copyWithHeader(process.Stdout(), wsNetConn, proto.Header{Type: proto.TypeStdout})
 			})
 			outputgroup.Go(func() error {
-				return copyWithHeader(process.Stderr(), wsNetConn, proto.Header{Type: proto.TypeStdout})
+				return copyWithHeader(process.Stderr(), wsNetConn, proto.Header{Type: proto.TypeStderr})
 			})
 
 			go func() {