TITLE: Identifyng objects in a category
QUESTION [5 upvotes]: Is there a general way of identifying objects in a category to produce a new category? Something like a quotient by a relation on objects. How would the morphisms behave?

REPLY [2 votes]: Yes, there is indeed. There is a notion of a generalized congruences, which is an equivalence relation on objects and sequences of morphisms. The notion was first introduced in  Generalized congruences - Epimorphisms in $\mathbf{Cat}$ by Marek A. Bednarczyk, Andrzej M. Borzyszkowski and Wieslaw Pawlowski (I found the notation in Emanuel Haucourts Categories of components and loop-free categories easier to read, though).
These generalized congruences appear naturally when calculating certain colimits in $\mathbf{Cat}$, in particular coequalizers, which is described in the first reference.
If you're interested in quotients by group actions, you might wanna look at Group actions on posets by Eric Babson and Dmitry Kozlov.