Equality in category theory seems poorly defined to me apologize if this doesn't make much sense, I am self-taught and often I am thinking about things completely wrong, but I am very lost right now.
When we consider some generalization of an idea[^1], say abstract vector spaces to R^2 or groups to symmetry or set theory to everything. There is usually a definition of equality of two objects.
For example, set equality is defined element wise in set theory, and equality in homotopy type theory is defined through paths[^2]
As I understand it, category theory is another type of these generalizations. But it generalizes a whole bunch of things into one theory. Like how it generalizes groups, posets, logic and a whole bunch of other things. But I have never seen anybody mention any notion of equality in category theory. And more so, it seems particularly useful. Consider the definition of monomorphisms and epimorphisms, they use equality, but equality has never been defined!
I'm not sure what to make of this, my instinct would be that it was an implicit part of the definition of a category, that it must have some sensible notion of equality. But in this case, how is equality defined on the category of small categories?
Please help, thank you :)
[^1] sorry if this isn't the right words to describe what I'm trying to say, hopefully the examples provide clarity on what I mean.
[^2] as well as definitionally
 A: I think it is misleading to equate(!) equivalence of categories with equality of categories. There is also the notion of isomorphism of categories.
Think about groups. Two groups can be different as actual sets with multiplication, but in the world of groups if they are isomorphic we often regard them as being the same. But actually this is really not quite what equality means since two groups can be isomorphic in many ways and it is only by choosing a specific isomorphism that you make them `equal'.
Similarly in category theory we can talk about two categories being isomorphic but for many purposes we can replace that with equivalent since that is often enough. Again, the similar situation with topological spaces is worth thinking about. Two spaces can be homeomorphic and a choice of homeomorphism means we can treat them as `equal'. But there is also the notion of homotopy equivalence which is frequently good enough: for example a disc and a point are homotopy equivalent, a circle and an annulus are, etc. In homotopy theory two homotopy equivalent spaces are essential the same as far as the sort of things that homotopy theorists do. In fact equivalent categories have homotopy equivalent classifying spaces so these notions are in some sense the same (equivalent?).
