Based on this diagram:
The Tasklist should be using a Rest API to communicate with the engine. I have a web application integrated with the engine. How can I make the Tasklist use the same engine I’m using? Which API is the Tasklist using?
The communication between the webapps and the REST-API is implemented in the camunda-bpm-sdk-js.
To find out what exactly is happening when the Tasklist makes a request, you can open the Development console of your browser (F12) and go to the Network tab. When you reload the page with this tab active, you see all requests the browser makes and can for example inspect which URL it is calling.