Is an over category of an essentially small category again essentially small? Is there an example of an non-essentially small over category $C\downarrow X$ of an essentially small category $C$?
 A: Edit. The previous argument had a flaw. Here is a more direct proof.
No such example exists.
Almost by definition, an essentially small category is locally small, so if $\mathcal{C}$ is essentially small, every slice category $\mathcal{C} / X$ is locally small. So we only need to show that there are only set-many isomorphism classes of objects in $\mathcal{C} / X$.
We know that $\mathcal{C}$ itself has only set-many isomorphism classes of objects: so let $\mathcal{O}$ be a set of objects of $\mathcal{C}$ such that every object of $\mathcal{C}$ is isomorphic to an object in $\mathcal{O}$. But objects in $\mathcal{C} / X$ are arrows of the form $p : E \to X$ in $\mathcal{C}$, and $E$ is isomorphic to some $E'$ in $\mathcal{O}$, so $p : E \to X$ is isomorphic to some $p' : E' \to X$ in $\mathcal{C} / X$. Thus, every object in $\mathcal{C} / X$ is isomorphic to an object in the set
$$\coprod_{E \in \mathcal{O}} \mathcal{C}(E, X)$$
and therefore, $\mathcal{C} / X$ is essentially small, as claimed.
