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Correct. That being said, there's nothing stopping you from emitting, say, HTML from such a function. But there's nothing really helping you either.

It's basically function calls in the cloud; a remote procedure call where the remote procedure is executing on their infrastructure.

I'm also guessing there's no session state or cookies?

We'll I guess there wouldn't be without a server. You'd have to use JWT or the like.

Or any server side store, maybe DynamoDB.

the functions are stateless, but you can access S3 and DynamoDB.