Q: Does the Mechanic verify the replacement part I think I need is what a actually need?

If you call out a mechanic to install a part that was diagnosed by another company then to answer your question is no. The mechanic that is to install a previously diagnosed parts cannot in most cases verify the part is going to correct the problem unless they were the one that diagnosed the problem. If you want a warranty on the repair then you should have the mechanic that diagnosed it to do the repairs. Otherwise you will not have a guarantee it will fix it.