2
$\begingroup$

From Categories for the Working Mathematician:

For small categories $A$, $B$, and $C$ establish a bijection

$$ \mathbf{Cat}(A \times B, C) \cong \mathbf{Cat}(A, C^B) $$

and show it natural in $A$, $B$, and $C$.

Here I'm assuming that $\mathbf{Cat}(A \times B, C)$ and $\mathbf{Cat}(A, C^B)$ denote the hom-sets $\text{hom}_\mathbf{Cat}(A \times B, C)$ and $\text{hom}_\mathbf{Cat}(A, C^B)$, respectively. So we are trying to establish a bijective function $f$ between these two sets.

Question: What does it mean to show that this bijection is "natural" in $A$, $B$, and $C$? I understand what a natural transformation is, but don't see the connection here to this question.

1 Answers 1

2

It should mean (I don't have access to the book at the moment) that, say, if you have a morphism $A\longrightarrow A'$, the diagram

$$\DeclareMathOperator{\Cat}{\bf Cat} \begin{matrix} \Cat(A'\times B,C)&\!\longrightarrow&\Cat(A',C^B)\\ \downarrow&&\downarrow\\% \Cat(A\times B,C)&\!\longrightarrow&\Cat(A,C^B) \end{matrix} $$ is commutative.

  • 0
    What Bernard suggests shows that the two functors are natural in A. You also must show they are natural in B and in C, using similar diagrams involving B' and B, and involving C' and C. – 2017-02-15