  We make some beginning observations about the category $\mathbb{E}\mathrm{q}$
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
$\mathbb{E}\mathrm{q}$, such as the category $\mathbb{E}\mathrm{q}(\Sigma^0_1)$
of computably enumerable equivalence relations (called ceers), the category
$\mathbb{E}\mathrm{q}(\Pi^0_1)$ of co-computably enumerable equivalence
relations, and the category $\mathbb{E}\mathrm{q}(\mathrm{Dark}^*)$ 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
$\mathbb{E}\mathrm{q}(\Sigma^0_1)$ the epimorphisms coincide with the onto
morphisms, but in $\mathbb{E}\mathrm{q}(\Pi^0_1)$ there are epimorphisms that
are not onto. Moreover, $\mathbb{E}\mathrm{q}$,
$\mathbb{E}\mathrm{q}(\Sigma^0_1)$, and $\mathbb{E}\mathrm{q}(\mathrm{Dark}^*)$
are closed under finite products, binary coproducts, and coequalizers, but we
give an example of two morphisms in $\mathbb{E}\mathrm{q}(\Pi^0_1)$ whose
coequalizer in $\mathbb{E}\mathrm{q}$ is not an object of
$\mathbb{E}\mathrm{q}(\Pi^0_1)$.
