Is the Natural Functor on the Quotient Category Ever Not the Identity Functor on Objects? Let $C$ be a category and $C/R$ its quotient category.
According to Wikipedia:

There is a natural quotient functor from C to C/R which sends each morphism to its equivalence class. This functor is bijective on objects and surjective on Hom-sets (i.e. it is a full functor).

Question: Notice the bolded above; why doesn't Wikipedia just state that this functor is the identity mapping on objects? Is it ever not the identity mapping (on objects)?

The short answer is "yes according to the definition provided in the link the object part of the functor is the identity", which is precisely the reason why the functor is bijective on objects.
I can only try to guess possible reasons on why the authors stressed the fact that the functor is bijective on objects, instead that simply saying that it is the identity on objects.
A possible reason could be that if you replace $\mathcal C/R$ with an isomorphic category you still get a get category which deserves the name of quotient category (one of category theory's motto is that we are interested in things up to, at least, isomorphisms) but such replacement does not ensure  that the objects of the new quotient category are the same as the objects of $\mathcal C$.
