Quotient category

Remarks

The compatibility condition makes the composition $[g] \circ [f] = [g \circ f]$ independent of the chosen representatives. The quotient functor $Q \colon \mathcal{C} \to \mathcal{C}/R$ is the identity on objects and sends $f$ to $[f]$. It is universal: a functor $G \colon \mathcal{C} \to \mathcal{D}$ with $G(f) = G(f')$ whenever $f \sim f'$ factors uniquely as $G = \bar{G} \circ Q$.

Example: view the monoid $(\Z, +)$ as a category with one object, whose morphisms are the integers and whose composition is addition. Let $m \sim n$ when $m - n$ is even. If $m \sim m'$ then $(k + m + l) - (k + m' + l) = m - m'$ is even, so $\sim$ is a congruence, and the quotient is the one-object category of the group $\Z/2\Z$.

Non-example: on the same category let $m \sim n$ when $m = n$ or $\{m, n\} = \{0, 1\}$. This is an equivalence relation, but $0 \sim 1$ while $0 + 1 = 1$ and $1 + 1 = 2$ are not related, so it is not compatible with composition and gives no quotient category.