$\require{AMScd}$ I am learning category theory and was thinking about how to rephrase the following fact:
Let $(X, \sim_X)$ and $(Y, \sim_Y)$ are topological spaces with equivalence relations and $f: X \rightarrow Y$. Suppose that $f$ is equivalence relation preserving (i.e. for each $a, b \in X$, $a \sim_X b$ implies that $f(a) \sim_Y f(b)$) and continuous. Then there exists a unique map $\bar{f}: X / {\sim_X} \rightarrow Y / {\sim_Y}$ such that $\bar{f}$ is continuous wrt the quotient topologies $\pi_Y \circ f = \bar{f} \circ \pi_X$, i.e. the following diagram commutes: \begin{CD} X @>{f}>> Y\\ @V{\pi_X}VV @VV{\pi_Y}V\\ X / {\sim_X} @>{\bar{f}}>> Y / {\sim_Y} \end{CD} where $\pi_X$ and $\pi_Y$ are the projection maps.
If we consider the category $\mathcal{C}$ with objects the topological spaces with an equivalence relation and morphisms the equivalence relation preserving continuous maps. Then the previous fact implies that existence of a functor $F : \mathcal{C} \rightarrow \textbf{Top}$ given by $F(X, \sim_X) = X/{\sim_X}$ and $F(f) = \bar{f}$.
Question: I was wondering about how to rephrase the uniqueness statement in terms of categories. The uniqueness goes like this: If $\pi_Y \circ f = g \circ \pi_X $, then \begin{CD} X @>{f}>> Y\\ @V{\pi_X}VV @VV{\pi_Y}V\\ X / {\sim_X} @>{\bar{f}}>> Y / {\sim_Y} \end{CD} implies that \begin{CD} X @>{\pi_X}>> X/{\sim_X}\\ @V{\pi_X}VV @VV{g}V\\ X / {\sim_X} @>{\bar{f}}>> Y / {\sim_Y} \end{CD} Since $\pi_X$ is a surjection, i.e. an epimorphism, $g = \bar{f}$
Is there some kind of category theoretic notion of what the defining diagram of $\bar{f}$ is?
Your $\pi_X: X \to X/\sim_X$ is the coequalizer of the pair of projection maps
$$E \overset{p_1}{\underset{p_2}{\rightrightarrows}} X$$
where $E = \{(x, y) \in X \times X|\; x \sim_X y\}$ is the equivalence relation, and $p_1(x, y) = x$ and $p_2(x, y) = y$.
The uniqueness is purely and simply due to the fact that $\pi_X$ is an epimorphism. All coequalizer maps are epimorphisms.