What is $\mathbf{Cat}$ in this exercise (II.5.1 p.44 Mac Lane Category theory)?

Question

I'm referring to exercise 1 of section 5 The category of all categories of Chapter II in Mac Lane's book Categories for working mathematicians (p. 44). The exercise asks to establish a bijection

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

for small categories $A,B$ and $C$ and show that it is natural in $A,B$ and $C$. But I don't have any idea of what $\mathbf{Cat}$ means. I guess he meant $\mathbf{Funct}$, the bifunctor presented some lines above. But then, in the next exercise he asks to establish natural ismorphisms

$$( A\times B)^C$$

and

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

and to compare the second with the bijection of exercise 1.

So, what does $\mathbf{Cat}$ mean? Is exercise 1 the same as the second part of exercise 2?

Thanks

Answer 1

As explained at the beginning of the "Hom-Sets" section, p.27, $C(a,b)$ is shorthand for $\operatorname{hom}_C(a,b)$, so the first exercise asks you to check that the set of functors from $A\times B$ to $C$ is in bijection with the set of functors from $A$ to the functor category $C^B$.

The second part of the second exercise wants you to regard $(-)^{(-)}$ as a functor $\operatorname{Cat^{op}} \times \operatorname{Cat} \to \operatorname{Cat}$, via the procedure described right above the exercises, and then check that the two functors are naturally isomorphic.

Answer 2

$\mathbf{Cat}$ is the category of small categories, so for small categories $A,B$, $\mathbf{Cat}(A,B)$ is the set of functors $A\to B$

(it's common to write $\mathscr{C}(A,B)$ for a category $\mathscr{C}$ and objects $A,B$ to denote the set of morphisms $A\to B$. In fact I'm pretty sure McLane uses and introduces that notation)

So you are asked to establish a natural isomorphism between the two functors.

Answer 3

In the paragraph above the exercise, he notes that Cat is the category of all small categories. (He defines the notation in chapter I, for instance, in the list of categories at the end of section I.2.) And in Section I.8, he notes that if $C$ is a category and $a$ and $b$ are objects, he uses $C(a, b)$ synonymously with $\hom(a, b)$. So this is a bijection of hom-sets (consisting of functors).