Are objects in the category Grp actually groups or isomorphism classes of groups? Is there a difference?

Question

I am starting to learn category theory. Because an isomorphism between groups A and B is just a function that is simultaneously a homomorphism from A to B and a homomorphism from B to A, it seems like that two groups being isomorphic doesn't mean that they are actually the same object in Grp. In general, it seems like having a morphism from A to B and a morphism from B to A is the same as having the natural equivalence in that category. So when we talk about Grp, are we talking about the collection of groups up to isomorphism, or are we using a different fundamental notion of "group"?

Answer 1

The definition I know for Grp has as objects groups and as morphisms homomorphisms of groups. This means two distinct but isomorphic groups are considered as two distinct objects.