If a category is locally small, any category equivalent to it is again locally small.
What I have: Let $D$ be a locally small category and $C$ be a category such that $C\simeq D$. Note that we have functors $F:C\to D$ and $G:\to C$ along with natural isomorphisms $\nu:id_C\simeq GF$ and $\alpha:FG\simeq id_D$. Now let $X,Y\in C$. We wish to show that there exists only a set's worth of morphisms between $X$ and $Y$. Note that $F$ is fully faithful. Therefore, the map $C(x,y)\to D(Fx,Fy)$ is bijective. Since $D$ is locally small, $D(Fx,Fy)$ is small enough to be a set. This means $C(x,y)$ is small enough to be a set. We conclude that $C$ is locally small.
Is this correct?
A functor defining an equivalence of categories is faithful and full.