Make sure that the sink vent is connected to the vent part of your assembly . anywhere above the top edge of your sink as a guide.
You are correct
Would i be right in thinking that any pressure from above re discharge would now have no bearing on the kitchen sink if there are two holes (via pipes) regarding the waste hole and the vent because there is no vacuum to be pushed or sucked from the kitchen pipe? becuase there is like a release loop created by the two holes.