Let $C$ be a category and $A\in\mathrm{ob}(C)$. A *relation* is a subobject $q:Q\hookrightarrow A^{\times 2}$ and the quotient is defined as the coequalizer
$$A/Q:=\mathrm{coeq}\left(Q\stackrel{q}{\hookrightarrow} A^{\times 2}\rightrightarrows A\right)$$
where the two maps are the two projections. Moreover, we can define what it means to be an *equivalence relation*: The inclusion $q$ should satisfy the following:

**Reflexivity:**Consider the diagonal $\Delta:A\to A^{\times 2}$. Then there should be a map $i:A\to Q$ such that $q\circ i=\Delta$.**Symmetry:**Consider the flip $t:A^{\times 2}\to A^{\times 2}$. Then there should be an automorphism $s:Q\to Q$ such that $t\circ q=q\circ s$.**Transitivity:**Consider the map $q\times_A q: Q\times_AQ\to A^{\times 2}\times_AA^{\times 2}$ and the outer projections $p:A^{\times 2}\times_AA^{\times 2}\to A^{\times 2}$. Then there should be a map $j:Q\times_A Q\to Q$ such that $p\circ (q\times_A q)=q\circ j$.

Given an arbitrary relation $q:Q\hookrightarrow A^{\times 2}$, we can define $\overline{Q}\stackrel{\overline{q}}{\hookrightarrow} A^{\times 2}$ as the inital equivalence relation having a morphism $i:Q\to \overline{Q}$ such that $\overline{q}\circ i=q$. I claim that the canonical map $$A/Q\to A/\overline{Q}$$ is an isomorphism. In $\mathbf{Set}$, this is clear as building the coequalizer is the same as quotienting out the spanned equivalence relation. Is this also true in other categories?