appsync lambda mutate and publish

I've seen how to create a lambda to perform an appsync mutation, invoking back ends such as dynamo. I've also seen examples of lambdas, which are appsync clients, which can publish events that clients subscribe to.

I've seen another example of this essentially working using a websocket AWS API endpoint with

conn = boto3.client("apigatewaymanagementapi"...
for chunk in s:
    conn.post_to_connection(**params)

I was hoping that the lambda would somehow have access to the appsync instance to receive a mutation request and then publish a result in a stream of messages to the client. Is this feasible, or do I just need to set up the lambda to send mutation events using its own separate connection to appsync using boto3? Instead of just returning a result, I want to start that stream of events going.

This is for a streaming AI use case with a python lambda:

• Client - sends mutation request to create a question with the same ID and subscribes to answerChunks coming back with that ID
• Lambda - receives the request, does "AI magic," - and then streams the response back in multiple "answerChunk" published events until the answer is finished.
• Client - aggregates the chunks into the output response display.