Suppose we have a set $S$ equipped with an equivalence relation $\sim$. The set of $\sim$ equivalence classes of $S$ is sometimes called 'the quotient set of $S$ by $\sim$'. This reminds me of other notions of quotient like a quotient group or a quotient vector space.
Can these be brought under the same umbrella in category theory as the name suggests? In particular, are there isomorphism theorems for equivalence relations just as there are for groups and vector spaces? Are any of these theorems the fundamental theorem of equivalence relations?