You can find a proof of this equivalence as Theorem IV.4.1 in Mac Lane's Categories for the Working Mathematician. Mac Lane does not explicitly mention the use of AC, but it is used at the start of the first full paragraph on page 94, where for each object $c$ in $C$ you choose an object $a_0$ and an isomorphism $c\to Sa$.
In fact, this result is equivalent to AC in ZF (assuming you state it only for small categories so it can be stated in ZF at all). To prove AC from this equivalence, let $I$ be any set of disjoint nonempty sets. Form a category $C$ whose collection of objects is $\bigcup I$ and in which there is a unique map $a\to b$ iff $a$ and $b$ are elements of the same element of $I$, and otherwise there are no maps $a\to b$. Let $D$ be the category whose objects are elements of $I$ and in which there are no morphisms besides identities. There is then a functor $F:C\to D$ which sends each object of $C$ to the element of $I$ which it is an element of. This functor is easily verified to be fully faithful and surjective on objects. If $F$ is an equivalence, then its inverse $G:D\to C$ will on objects be exactly a choice function on $I$.
(If you work instead in something like NBG without choice and state the result for large categories, then a similar argument shows that it is equivalent to global choice.)
There are many other statements which occur naturally in category theory which are equivalent to or require AC. The most simple example is the assertion that every epimorphism in the category of sets splits. Concretely, this just means that if $f:X\to Y$ is a surjection of sets then there exists $g:Y\to X$ such that $f(g(y))=y$ for all $y\in Y$. This is easily seen to be equivalent to AC, since such a $g$ is equivalent to choosing an element of each fiber of $f$.
More generally, many statements involving epimorphisms or surjections require choice. The characterization of equivalences of categories is one such statement: the axiom of choice is needed exactly because you need to make choices that witness the essential surjectivity of the functor. Another source of such statements is when talking about projective objects: in order to prove objects are projective, you frequently need the axiom of choice.