If $R\subset C$ is a vector subspace of $C$, and $F: C \to D$ is a linear map, then we have a natural linear map $C/R\to D/F(R)$. I was wondering if this can be also generalized to categorical settings.
For example, let $\mathscr F:\mathscr C\to\mathscr D$ be a functor and let $\mathcal R$ be a congruence relation on $\mathscr C$ which gives rise to a **quotient category** $\mathscr C/\mathcal R$.

Question 1: Under what conditions can we get a congruence relation $\mathcal S$ which somehow represents ''$\mathscr F(\mathcal R)$'' so that whenever $f\sim_{\mathcal R}g$, we have $\mathscr F(f)\sim_{\mathcal S} \mathscr F(g)$? If so, do we have a natural functor like $\mathscr C/\mathcal R\to \mathscr D/\mathscr F(\mathcal R)$?

**EDIT**: It seems that there is a nice situation where a sort of `pushforward' could be reasonable:

For simplicity, let's assume everything is **strict**.
Take a $\mathscr S$-indexed category, namely, a 2-functor $\pmb C:\mathscr S^{op}\to \mathsf{Cat}$. Explicitly, we have a category $\pmb C(I)$ for each object $I$ of $\mathscr S$ and a functor $f^*:\pmb C(I) \to \pmb C(J)$ for every arrow $f:J\to I$ satisfying $(\mathrm{id}_I)^*=\mathrm{id}_{\pmb C(I)}$ and $(g\circ f)^*=f^*g^*$.
Then it seems that every congruence relation $\mathcal R$ on $\mathscr S$ gives rise to a `congruence relation' for $\pmb C$ by requiring $f^*\approx g^*$ whenever $f\sim_{\mathcal R} g$.
Then this relation '$\approx$' is sort of analogous to a congruence relation in that if $f_1^*\approx f_2^*$ and $g_1^*\approx g_2^*$, i.e. $f_1\sim_{\mathcal R} g_2$ and $g_1\sim_{\mathcal R} g_2$, then by definition $g_1\circ f_1\sim_{\mathcal R} g_2\circ f_2$ and therefore we see $f_1^*g_1^*=(g_1\circ f_1)^*\approx (g_2\circ f_2)^* =f_2^*g_2^*$

So, it is very interesting to ask:

Question 2: How do we develop 'quotients' of indexed categories?

For example, suppose $\pmb C:\mathscr S^{op}\to \mathsf {Cat}$ is an indexed category. As we discussed above, a reasonable quotient of $\pmb C$ might come from a quotient of $\mathscr S$. In view of another post, suppose there was indeed a reasonable definition of quotient indexed category, what concept of fibered categories was related to it under the Grothendieck construction?