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 an arbitrary abelian category. How do we resolve this?


(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 $\mathcal{A}$?



Universal Objects in Big Categories
In ZFC, a class is given by a class formula $\phi$, that is, a formula expressed in the language of ZFC. For example, the notion of group can be formalized by such a formula $\phi$. The notion of group homomorphism can also be so formalized, say by $\psi$. So in principle there is no problem in being able to write down a formula in ZFC whose abbreviation is 


$$\exists 1 \ \forall G \ \phi(G) \Rightarrow \exists ! f \ \psi(f) \wedge dom(f) = G \wedge cod(f) = 1$$ 


and this is what you're after. 


"Unbounded" quantification takes place all the time in ZFC. For example, the pairing axiom involves the formation of a set $\{x, y\}$ for any two sets $x, y$, and this "any" involves quantification over the class of sets. The axioms of set theory should be formulated to avoid completely unbounded instances of say the comprehension or separation axiom, as we know from the classical set-theoretic paradoxes, but all this is well worked out. 


