diff --git a/pipe/pipeline.go b/pipe/pipeline.go
index d237461..44fedd1 100644
--- a/pipe/pipeline.go
+++ b/pipe/pipeline.go
@@ -286,9 +286,6 @@ func (p *Pipeline) Start(ctx context.Context) error {
 	// Store the stages in the joiners, and verify that the stages'
 	// requirements are well-formed:
 	for i, s := range p.stages {
-		stageJoiners[i].nextStage = s
-		stageJoiners[i+1].prevStage = s
-
 		// Make sure that the stage's requirements are well-formed:
 		requirements := s.Requirements()
 		if err := requirements.Stdin.Validate(); err != nil {
@@ -297,21 +294,25 @@ func (p *Pipeline) Start(ctx context.Context) error {
 		if err := requirements.Stdout.Validate(); err != nil {
 			return fmt.Errorf("stdout: %w", err)
 		}
+
+		stageJoiners[i].nextStage = s
+		stageJoiners[i].nextStageReq = requirements
+		stageJoiners[i+1].prevStage = s
+		stageJoiners[i+1].prevStageReq = requirements
 	}
 
-	// Create the "inner" pipes (i.e, all but the first and last
-	// `stageJoiners`):
-	for i := 1; i < len(stageJoiners)-1; i++ {
-		if err := stageJoiners[i].createPipe(); err != nil {
+	// Check that each of the stages' requirements are satisfiable:
+	for i := range stageJoiners {
+		if err := stageJoiners[i].validate(); err != nil {
 			closePipes()
 			return err
 		}
 	}
 
-	// Check that each of the stages' requirements are compatible with
-	// the pipes that we have created for them:
-	for i := range stageJoiners {
-		if err := stageJoiners[i].validate(); err != nil {
+	// Create the "inner" pipes (i.e, all but the first and last
+	// `stageJoiners`):
+	for i := 1; i < len(stageJoiners)-1; i++ {
+		if err := stageJoiners[i].createPipe(); err != nil {
 			closePipes()
 			return err
 		}
diff --git a/pipe/stage_joiner.go b/pipe/stage_joiner.go
index 24fb789..03569cb 100644
--- a/pipe/stage_joiner.go
+++ b/pipe/stage_joiner.go
@@ -43,6 +43,11 @@ type stageJoiner struct {
 	// prevStage holds the stage that needs to write to the pipe.
 	prevStage Stage
 
+	// prevStageReq caches `prevStage.Requirements()` so that it
+	// doesn't have to be recomputed. It is the zero value if
+	// `prevStage` is nil.
+	prevStageReq StageRequirements
+
 	// prevStdout will be used as the stdout of `prevStage`. It is
 	// usually the "write" end of the `(nextStdin, prevStdout)` pipe
 	// pair, with the connected pipe ends in the same `stageJoiner`
@@ -52,6 +57,11 @@ type stageJoiner struct {
 	// nextStage holds the stage that needs to read from the pipe.
 	nextStage Stage
 
+	// nextStageReq caches `nextStage.Requirements()` so that it
+	// doesn't have to be recomputed. It is the zero value if
+	// `nextStage` is nil.
+	nextStageReq StageRequirements
+
 	// nextStdin will be used as the stdin of `nextStage`. It is
 	// usually the "read" end of the `(nextStdin, prevStdout)` pipe
 	// pair.
@@ -61,13 +71,8 @@ type stageJoiner struct {
 // needFilePipe returns `true` if the pipe that joins the two adjacent
 // stages should be an `os.Pipe()` rather than an `io.Pipe()`.
 func (sj *stageJoiner) needFilePipe() bool {
-	if sj.prevStage.Requirements().Stdout == StreamPreferFile {
-		return true
-	}
-	if sj.nextStage.Requirements().Stdin == StreamPreferFile {
-		return true
-	}
-	return false
+	return sj.prevStageReq.Stdout == StreamPreferFile ||
+		sj.nextStageReq.Stdin == StreamPreferFile
 }
 
 func (sj *stageJoiner) createPipe() error {
@@ -100,26 +105,28 @@ func (sj *stageJoiner) closePipe() error {
 	)
 }
 
-// validate verifies that `sj.prevStdout` and `sj.nextStdin` are
-// suitable for the adjacent stages, in particular that no pipe is
-// created if the stage requirements are `StreamForbidden`.
+// validate verifies that the adjacent stages' stream requirements are
+// satisfiable, in particular that a stage that forbids its stdin or
+// stdout is not connected to anything.
 func (sj *stageJoiner) validate() error {
-	if sj.prevStage != nil {
-		stdoutRequirements := sj.prevStage.Requirements().Stdout
-		if stdoutRequirements == StreamForbidden && sj.prevStdout != nil {
-			return fmt.Errorf(
-				"stage %q forbids stdout, but stdout is connected", sj.prevStage.Name(),
-			)
-		}
+	// `prevStage`'s stdout is connected if there is a `nextStage` to
+	// consume it (in which case an inner pipe will be created) or if
+	// a stream (`p.stdout`) has already been stored in `prevStdout`.
+	if sj.prevStage != nil && sj.prevStageReq.Stdout == StreamForbidden &&
+		(sj.nextStage != nil || sj.prevStdout != nil) {
+		return fmt.Errorf(
+			"stage %q forbids stdout, but stdout is connected", sj.prevStage.Name(),
+		)
 	}
 
-	if sj.nextStage != nil {
-		stdinRequirements := sj.nextStage.Requirements().Stdin
-		if stdinRequirements == StreamForbidden && sj.nextStdin != nil {
-			return fmt.Errorf(
-				"stage %q forbids stdin, but stdin is connected", sj.nextStage.Name(),
-			)
-		}
+	// `nextStage`'s stdin is connected if there is a `prevStage` to
+	// produce it (in which case an inner pipe will be created) or if
+	// a stream (`p.stdin`) has already been stored in `nextStdin`.
+	if sj.nextStage != nil && sj.nextStageReq.Stdin == StreamForbidden &&
+		(sj.prevStage != nil || sj.nextStdin != nil) {
+		return fmt.Errorf(
+			"stage %q forbids stdin, but stdin is connected", sj.nextStage.Name(),
+		)
 	}
 
 	return nil