auto vent/air gap guidance

during a recent kitchen remodel we had some plumbing roughed in. the plumber was out of town and sent his assistant who did things a little different including plumbing for an autovent instead of tapping into the cast iron line as was planned for by our original plumber but he thought that was too tedious and old school. my question is this....does the assembly have to be above countertop/sink? if i am correct i think it has to be above the overflow line of the sink, but I'm having a hard time explaining to the significant other who wants no visible plumbing.

Most places require an air gap for the dishwasher.
I'm not sure by the picture above what he intends to do.

we have no idea WHY he has the second pipe in the wall, BUT if "my assistant" decided doing something the "right way" was "tedious and old school", he would find himself on the operating end of a shovel tomorrow. And stay there until he decided maybe it was NOT so tedious and old school.

i believe the upper non threaded 1.5" elbow is for the auto vent it was originally vertical but with .75" foam board insulation and .5" sheet rock the wall>cabinet>pipe clearance was an issue and since all this was roughed in we minimized the vertical run of the 'Y' fitting that the vent gap was to fit on. there was no way we could have completed everything and lifted the sink base over that set up. i think he was expecting us to have the plumbing come up through the bottom of the cabinet VS through the back as it is now. the new problem is how and where to attach the thing without having to cut a hole in the countertop (not much room), which has me wondering if there is another way, or can it remain under the sink in the horizontal position as shown with that upper 90deg. fitting. if installation calls for it to sit on sink or countertop as in the diagram displayed with Terry's response, then thats the way i will go. if not, I'm open to suggestions.