There is a natural quotient functor from C to C/R which sends each morphism to its equivalence class.
The concept of an additive congruence relation is equivalent to the concept of a two-sided ideal of morphisms: for any two objects X and Y we are given an additive subgroup I(X,Y) of HomC(X, Y) such that for all f ∈ I(X,Y), g ∈ HomC(Y, Z) and h∈ HomC(W, X), we have gf ∈ I(X,Z) and fh ∈ I(W,Y).
Two morphisms in HomC(X, Y) are congruent iff their difference is in I(X,Y).
Every unital ring may be viewed as an additive category with a single object, and the quotient of additive categories defined above coincides in this case with the notion of a quotient ring modulo a two-sided ideal.
This tends to increase the number of morphisms between objects, rather than decrease it as in the case of quotient categories.