The category of equivalence relations

We make some beginning observations about the category $eeq$ of equivalence relations on the set of natural numbers, where a morphism between two equivalence relations $R,S$ is a mapping from the set of $R$-equivalence classes to that of $S$-equivalence classes, which is induced by a computable function. We also consider some full subcategories of $eeq$, such as the category $eeq(Sigma^0_1)$ of computably enumerable equivalence relations (called emph{ceers}), the category $eeq(Pi^0_1)$ of co-computably enumerable equivalence relations, and the category $eeq(dark^*)$ whose objects are the so-called dark ceers plus the ceers with finitely many equivalence classes. Although in all these categories the monomorphisms coincide with the injective morphisms, we show that in $eeq(Sigma^0_1)$ the epimorphisms coincide with the onto morphisms, but in $eeq(Pi^0_1)$ there are epimorphisms that are not onto. Moreover, $eeq$, $eeq(Sigma^0_1)$, and $eeq(dark^*)$ are closed under finite products, binary coproducts, and coequalizers. On the other hand, we show that $eeq(Pi^0_1)$ does not always have coequalizers