dishwasher drainage

Looking to figure out how I can join these to the drainage that goes to the sewer. I guess I need to get a drainage trap like this , but not sure what else is needed to join the black pipe (shower) and grey pipe (sink and diswasher) The drainage pipe goes down into an elbow into the sewer...

My Hotpoint HSFO 3T223 started displaying the F03 error code which indicates a drainage issue. I've began by cleaning out the internal filters which had barely anything inside, I then disconnected from the wall and the waste under my sink and drained the waste pipe out manually. This didn't work...