Unlike the category of small categories (with functors), the category of small semigroupoids (like categories except that identity arrows are not required to exist, with semifunctors) is not cartesian closed. One can define "natural transformations" between semifunctors in the obvious way, as well as the "semifunctor semigroupoid", but that will not generally represent an exponential object.
When is a semigroupoid $S$ exponentiable in $\mathbf{SemiGroupoid}$, the category of small semigroupoids?
Note that semigroupoids work differently from categories. In particular, objects correspond to semifunctors from the semigroupoid with one object and no arrows, while arrows correspond to semifunctors from the semigroupoid with two objects $0$ and $1$ and an arrow $0 \to 1$, but not an arrow $1 \to 0$ or self-loops. In contrast, semifunctors from the category (which is a fortiori also a semigroupoid) with one object and an identity arrow correspond to idempotent arrows from an object to itself.
In particular, if $T^S$ is an exponential object in $\mathbf{SemiGroupoid}$, then its objects and arrows must correspond as follows:
- The objects in $T^S$ must be in bijection with maps from objects of $S$ to objects of $T$, ignoring the arrows.
- The arrows in $T^S$ from $F_1$ to $F_2$ must be in bijection with maps from arrows in $S$ to arrows in $T$ that send each arrow $A_1 \to A_2$ in $S$ to an arrow $F_1(A_1) \to F_2(A_2)$ in $T$.