TITLE: Equivalence of categories implying, under some special hypothesis, isomorphism of categories
QUESTION [6 upvotes]: If two categories $C$ and $D$ are equivalent, this implies that there's a bijection between the isomorphism classes of objects of $C$ and the isomorphism classes of objects of $D$. If we furthermore assume that for each isomorphism class of objects of $C$, the corresponding isomorphism class of objects of $D$ has the same cardinality, does it follow that $C$ is isomorphic to $D$?

REPLY [3 votes]: Yes.
Let $[X]$ denote the isomorphism class of $X$, and let $F : C \to D$ be a functor giving rise to an equivalence of categories (that is, fully faithful and essentially surjective). Suppose that $\forall X \in C |[X]| = |[F(X)]|$.
Note that if $[X] = [Y]$ then $[F(X)] = [F(Y)]$.
Choose, for each isomorphism class $[X]$, a bijection $b_{[X]} : [X] \to [F(X)]$.
Choose, for each $X$, an isomorphism $i_X : F(X) \to b_{[X]}(X)$.
Define a new functor $G : C \to D$ which acts on objects by $G(X) = b_{[X]}(X)$ and which acts on arrows $f : A \to B$ by $G(f) = i_B \circ F(f) \circ i_A^{-1}$.
Then $G$ is an isomorphism of categories since it is a bijection on objects and is fully faithful.