Is the Natural Functor on the Quotient Category Ever Not the Identity Functor on Objects?

Question

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)?

Answer 1

The most likely answer is that the author found this the most natural language to express this.

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That said, there are reasons not to insist on it being an identity map — even with the specific construction of the quotient category you have in mind. For example:

• There is an "arrow only" definition of category, in which the word "object" means "identity arrow". The quotient functor map constructed here is not the functor when applied to identity arrows.
• Following various lines of thought of category theory, the notion of 'object' should be so that every object remembers what category it is a member of. Thus, it simply doesn't make sense to suggest that a functor between different categories is the identity on objects.

One of the first examples one often sees in the spirit of the second bullet is the distinction between a morphism of sets (which remembers its domain and codomain) and the set of points of its graph (which only remembers the domain).

Answer 2

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 $\mathcal C$.