The adapter has a couple of generic request handlers that you can use. The most generic is SpringBootStreamHandler, which uses a Jackson ObjectMapper provided by Spring Boot to serialize and deserialize the objects in the function. There is also a SpringBootRequestHandler which you can extend, and provide the input and output types as type parameters (enabling AWS to inspect the class and do the JSON conversions itself).
If your app has more than one @Bean of type Function etc. then you can choose the one to use by configuring function.name (e.g. as FUNCTION_NAME environment variable in AWS). The functions are extracted from the Spring Cloud FunctionCatalog (searching first for Function then Consumer and finally Supplier).
You don’t need the Spring Cloud Function Web or Stream adapter at runtime in Lambda, so you might need to exclude those before you create the JAR you send to AWS. A Lambda application has to be shaded, but a Spring Boot standalone application does not, so you can run the same app using 2 separate jars (as per the sample here). The sample app creates 2 jar files, one with an aws classifier for deploying in Lambda, and one executable (thin) jar that includes spring-cloud-function-web at runtime. Spring Cloud Function will try and locate a "main class" for you from the JAR file manifest, using the Start-Class attribute (which will be added for you by the Spring Boot tooling if you use the starter parent). If there is no Start-Class in your manifest you can use an environment variable MAIN_CLASS when you deploy the function to AWS.