API Gateway doesn't invoke the Lambda function using "the network" as you have defined it (VPC in this case).
When you create an integration between API Gateway and a Lambda function you authorise API Gateway to invoke the function; and you provide permissions in Lambda for API Gateway to invoke it (it's a two-way operation).
"Traffic" (in this case, the request from APi Gateway to Lambda and the response from Lambda to API Gateway) is carried across the AWS services control plane so you don't need to provide a network path for that to happen.
Any network traffic generated by the Lambda function (for example, accessing a database or calling another AWS service) will (in your case) will originate from the network interface Lambda has on your private subnet.
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