I have a AWS API Gateway with websocket connection. And there is a lambda (LA) which is called by this API gateway and it sends a request to another service along with the websocket connection ID. After that it exists.
There is another lambda LB which is listening on some backend data store. The data store service will call LB if there is any update event. The event includes websocket conneciton ID. And I'd like to let LB send an update event to API gateway which forward this event to clients over websocket connection.
This is a communication mode in my mind. My question is whether API gateway supports such mode. If yes, how does LB send response event to API gateway and to clients?
The communication looks like below screenshot. The question is whether it works? how the response lambda send event to API gateway?
I have added the Integration response as below. There are Integration Request and Integration Response. But I can only set up one lambda for both of them. How can I make different lambda for request and response?

