Show that any category that is equivalent to a locally small category is locally small

Question

I'm studying Category Theory in Context, and I'm stuck on this seemingly easy problem. Hints would be appreciated.

My attempt:

Let $C$ be a locally small category, and let $C \cong D$. (i.e. $C$ and $D$ are equivalent.)

We note that there exists fully faithful functor $F: C \rightarrow D$.

Fix $x, y \in D$.

Here's where I'm stuck. If we could find $x', y' \in C$ such that $F(x') = x$ and $F(y') = y$, we would be done since the map sending $C(x',y')$ to $D(F(x'), F(y'))$ is bijective and $C(x',y')$ is a set.

But we are not guaranteed to find such $x'$ and $y'$.

We also know that $F$ is essentially surjective, but that doesn't seem to help either.

I do realize there's a similar question here: If a category is locally small, any category equivalent to it is again locally small. but it doesn't seem to address my concern.

Thanks for your help!

Answer 1

Essential surjectivity of $F$ does help. Indeed, if $x \cong F(x')$ and $y\cong F(y')$, then $D(x,y)\cong D(F(x'), F(y'))$, and so you may then apply your reasoning to get that this is $\cong C(x',y')$.