One knows that many models of set theory exist. In topos theory,"the" category of sets is to play the role of the point. Since many models of set theory are around, I believe one of the following to be true.

There is one category of sets and the model determines what is true about the category of sets.

There are many different categories that can be called the category of sets, one for each model.

The question is: is 1 or 2 correct? Perhaps neither one is correct. In either case,what would a good reference be?