Let $G$ be a group, $\mathcal{N}$ the collection of normal subgroups of $G$ and $\mathcal{C}$ the collection congruence relations on $G$. In my group theory class we showed there is a bijective correspondence between $\mathcal{N}$ and $\mathcal{C}$ via $$N \mapsto \left[ a\sim b \iff a^{-1}b \in N \right]$$ and $${\sim} \mapsto [1]$$
Can this be extended to an equivalence of categories?
The difficulty I’m having is understanding the morphisms. I tried taking $\hom(\mathcal{N})$ to consist of group homorphisms $\phi : M \to N$. And $\hom(\mathcal{C})$ to consist of maps $f:(G,\sim_1) \to (G,\sim_2)$ such that $a \sim_1 b$ implies $f(a) \sim_2 f(b)$. But I don't see where these morphisms should be mapped to.
It appears to me that you can set
$$\operatorname{hom}(\sim_1,\sim_2)=\begin{cases} \{\langle\sim_1,\sim_2\rangle\},&\text{if }{\sim_1}\supseteq{\sim_2}\\ \varnothing,&\text{otherwise}\;, \end{cases}$$
and
$$\operatorname{hom}(N_1,N_2)=\begin{cases} \operatorname{id}_{N_1},&\text{if }N_1\le N_2\\ \varnothing,&\text{otherwise}\;, \end{cases}$$
much as is done in turning a partial order into a category. I’m not sure that it’s at all interesting, but it does seem to work.