There are two common ways to define a category:
As a collection of objects, together with for each pair of objects, a collection of morphisms, together with identities and composition maps , satisfying axioms.
In logical terms, the first is formulated as a -sorted theory in first-order logic, while the second is a dependently typed theory; the usual way of interpreting any dependently typed theory as an independently sorted theory turns the latter into the former.
However, there is a third way of defining a category which uses only one collection (representing the collection of morphisms) and thus is formulated as an untyped (or -sorted) first-order theory. Note that this is not the usual way of replacing sorts with predicates, but instead a slightly clever trick. The basic idea is that an object can be identified with its identity morphism. This reformulation is occasionally useful, but mostly for technical reasons.
The first two axioms say that and are idempotent endofunctions on which have the same image. The elements of their common image (the such that , or equivalently ) are called identities or objects. Once that is done, the rest of the identification is straightforward.
A functor between single-sorted categories is just a function such that , , and whenever is defined (which, by the first two axioms of a functor and axiom (3) of a category, implies that is defined).
Finally, a natural transformation between functors of single-sorted categories is a function such that , , and whenever is defined (which implies that both composites in this identity are defined). Note that while a natural transformation is ordinarily defined to consist of a component only when is an object, this definition supplies a component to each morphism. In terms of the usual definition, the component of at a morphism is the diagonal of the corresponding naturality square.
A monoid is a single-sorted category in which is a constant function (hence so is , and they are equal). This works up to isomorphism of categories, not merely equivalence, so single-sorted categories may seem to be a more direct oidification of monoids than the usual categories.
The usual definition of an internal category is two-sorted, but the one-sorted definition can also be interpreted internally. While the usual notion of internal category requires the ambient category only to have pullbacks, the one-sorted version appears to require one to make sense of an “internal partial binary operation.” However, since in this case the domain of is specified explicitly in the definition, one can just require to be an ordinary morphism whose domain is the pullback of and ; thus only pullbacks are required for the single-sorted definition as well.
It is easy to see that any internal two-sorted category gives an internal one-sorted category (consider the object of arrows). The converse is true as long as the ambient category has split idempotents, for then given an internal one-sorted category we can split either or to obtain an object of objects. In general, however, the two concepts are not equivalent.
There exist similar single-sorted definitions of -categories and ∞-categories. The single sort in the definition of -category is the set of -morphisms, but you can also think of this as the union (over all ) of the sets of -morphisms, as long as you identify each -morphism (for ) with its identity -morphism. In the the definition of -category, there is no notion of -morphism to take care of everything at once, but the single sort can still be understood as this union (now over all ).