Please explain Mac Lane notation on page 48

Question

On page 47-48 of Mac Lane's book on Category theory, he uses the notation $\mathbb C^{d_0}$ to denote the functor $\mathbb C^2\to \mathbb C$, defined by $(a:A\to B)\mapsto A$, which takes an arrow to its domain. Can somebody please explain that notation. Mac Lane himself refers to "the end of the last section", but I didn't find there a clear definition.

Answer 1

Let $\mathcal{C}$ be a category and $\mathbb{D}$ and $\mathbb{E}$ (small) categories. Let $F : \mathbb{D}\to\mathbb{E}$ be a functor. $\mathcal{C}^F : \mathcal{C}^{\mathbb{E}}\to\mathcal{C}^{\mathbb{D}}$ and is defined by $G \mapsto G \circ F$. In this scenario Mac Lane is using, $d_0 : \mathbf1 \to \mathbf 2$ and so $\mathcal{C}^{d_0} : \mathcal{C}^{\mathbf{2}}\to\mathcal{C}^{\mathbf{1}}$. Mac Lane then tacitly uses the equivalence $\mathcal{C}\simeq\mathcal{C}^{\mathbf{1}}$.

An arrow in $\mathcal{C}$ can be identified a functor $\mathbf{2}\to\mathcal{C}$ and an object of $\mathcal{C}$ can be identified with a functor $\mathbf{1}\to\mathcal{C}$. You should work out the details of this if you haven't yet.

More generally, this exponential notation is commonly used in Cartesian closed (or even more generally, monoidally closed) categories.