Let $\mathcal{A}$ be a category whose collection of object forms a proper class. Then, to be able to formulate the concept of a terminal object in $\mathcal{A}$, do we have to leave ZFC? In other words, can we formulate the concept of a terminal object in $\mathcal{A}$ inside ZFC. If yes, how could we do it?

For example, in the category $\mathcal{Sets}$ of all sets. A singleton is a terminal object in $\mathcal{Sets}$, which could be checked by hand. But if we want to actually define what a terminal object is, then we have to quantify over all Sets, which is not a set. So, the definition is ill-formed inside ZFC?

A similar issue arises when we talk about things like injective objects, projective objects etc. of $\mathcal{Ab}$.

(I must be very confused about this basic issue; any help to resolve this will be greatly appreciated)

EDIT 1: What happens if in $\mathcal{A}$, $\hom(A,B)$ may also be proper classes where $A$ and $B$ are arbitrary objects of $\mathcal{A}$?