diff --git a/spring-cloud-function-context/src/main/java/org/springframework/cloud/function/context/catalog/SimpleFunctionRegistry.java b/spring-cloud-function-context/src/main/java/org/springframework/cloud/function/context/catalog/SimpleFunctionRegistry.java
index e470d1b5e..c2233d09e 100644
--- a/spring-cloud-function-context/src/main/java/org/springframework/cloud/function/context/catalog/SimpleFunctionRegistry.java
+++ b/spring-cloud-function-context/src/main/java/org/springframework/cloud/function/context/catalog/SimpleFunctionRegistry.java
@@ -636,6 +636,7 @@ public class SimpleFunctionRegistry implements FunctionRegistry {
 		@Override
 		public <V> Function<Object, V> andThen(Function<? super Object, ? extends V> after) {
 			Assert.isTrue(after instanceof FunctionInvocationWrapper, "Composed function must be an instanceof FunctionInvocationWrapper.");
+
 			if (FunctionTypeUtils.isMultipleArgumentType(this.inputType)
 					|| FunctionTypeUtils.isMultipleArgumentType(this.outputType)
 					|| FunctionTypeUtils.isMultipleArgumentType(((FunctionInvocationWrapper) after).inputType)
@@ -643,6 +644,8 @@ public class SimpleFunctionRegistry implements FunctionRegistry {
 				throw new UnsupportedOperationException("Composition of functions with multiple arguments is not supported at the moment");
 			}
 
+			this.setSkipOutputConversion(true);
+			((FunctionInvocationWrapper) after).setSkipOutputConversion(true);
 			Function rawComposedFunction = v -> ((FunctionInvocationWrapper) after).doApply(doApply(v));
 
 			FunctionInvocationWrapper afterWrapper = (FunctionInvocationWrapper) after;
diff --git a/spring-cloud-function-context/src/test/java/org/springframework/cloud/function/context/catalog/SimpleFunctionRegistryTests.java b/spring-cloud-function-context/src/test/java/org/springframework/cloud/function/context/catalog/SimpleFunctionRegistryTests.java
index db2a902c9..fe012044a 100644
--- a/spring-cloud-function-context/src/test/java/org/springframework/cloud/function/context/catalog/SimpleFunctionRegistryTests.java
+++ b/spring-cloud-function-context/src/test/java/org/springframework/cloud/function/context/catalog/SimpleFunctionRegistryTests.java
@@ -59,6 +59,7 @@ import org.springframework.cloud.function.json.JacksonMapper;
 import org.springframework.cloud.function.json.JsonMapper;
 import org.springframework.context.ApplicationContext;
 import org.springframework.context.annotation.Bean;
+import org.springframework.context.annotation.Configuration;
 import org.springframework.core.ResolvableType;
 import org.springframework.core.convert.ConversionService;
 import org.springframework.core.convert.support.DefaultConversionService;
@@ -550,6 +551,14 @@ public class SimpleFunctionRegistryTests {
 		assertThat(FunctionTypeUtils.isMono(function.getOutputType()));
 	}
 
+	@SuppressWarnings({ "rawtypes", "unchecked" })
+	@Test
+	public void testHeaderPropagationInComposedFunction() {
+		FunctionCatalog catalog = this.configureCatalog(GH_1063_Configuration.class);
+		Consumer function = catalog.lookup("uppercase|reverse|print");
+		function.accept("hello");
+	}
+
 	@Test
 	public void testFunctionCompositionWithReactiveSupplierAndConsumer() {
 		SimpleFunctionRegistry catalog = new SimpleFunctionRegistry(this.conversionService, this.messageConverter,
@@ -820,4 +829,24 @@ public class SimpleFunctionRegistryTests {
 		}
 	}
 
+	@EnableAutoConfiguration
+	@Configuration
+	public static class GH_1063_Configuration {
+
+		@Bean
+		Function<String, Message<String>> uppercase() {
+			return input -> MessageBuilder.withPayload(input).setHeader("FOO", "BAR").build();
+		}
+
+		@Bean
+		Function<String, String> reverse() {
+			return payload -> new StringBuilder(payload).reverse().toString();
+		}
+
+		@Bean
+		Consumer<Message<String>> print() {
+			return msg -> assertThat(msg.getHeaders().get("FOO")).isEqualTo("BAR");
+		}
+	}
+
 }