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
Best Answer
$\mathbf{Cat}$ 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.