Check Valve
 

I think its also called the check-and-cut valve? I am replacing my emission system and I was wondering how important this is to the system and if I can remove it for emission testing without consequences.

It runs from the main cat converter to the engine as a small long tube. On page 165 of the Haynes Manual split air goes to the main converter from this valve.

I did a search, but those other check valves refer to the one that circulate the engine and not the cat.

That check valve keeps exhaust gases on the correct side. The port air path has one behind the ACV as well. If either of these is defective, hot exhaust can bake your air pump.