Summary: 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 \(R\) and \(S\) 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 \(\mathbb{E}\mathrm{q}\left({\Sigma}_1^0\right)\) of computably enumerable equivalence relations (called ceers), the category \(\mathbb{E}\mathrm{q}\left({\Pi}_1^0\right)\) of co-computably enumerable equivalence relations, and the category \(\mathbb{E}\mathrm{q}(\text{Dark}^*)\) 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 \(\mathbb{E}\mathrm{q}\left({\Sigma}_1^0\right)\) the epimorphisms coincide with the onto morphisms, but in \(\mathbb{E}\mathrm{q}\left({\Pi}_1^0\right)\) there are epimorphisms that are not onto. Moreover, \( \mathbb{E}\mathrm{q}\), \( \mathbb{E}\mathrm{q}\left({\Sigma}_1^0\right)\), and \(\mathbb{E}\mathrm{q}(\text{Dark}^*)\) are closed under finite products, binary coproducts, and coequalizers, but we give an example of two morphisms in \(\mathbb{E}\mathrm{q}\left({\Pi}_1^0\right)\) whose coequalizer in \(\mathbb{E}\mathrm{q}\) is not an object of \(\mathbb{E}\mathrm{q}\left({\Pi}_1^0\right)\).