Let $\mathscr{C}$ be a locally small category. Then $f: c \to c'$ monic is equivalent to $$ f_*: \text{Hom}(x,c) \to \text{Hom}(x,c'), \; (l: x \to c) \mapsto f \circ l $$ being injectiv for all $x \in \mathscr{C}$.
My question: Is there a way to make sense of such an equivalence on categories which are not locally small? Might there be a way to speak of injective maps between classes?
Background: I always took this ($f$ being monic $\Leftrightarrow f_*$ injective) as the definition of a monomorphism since its seems to be both very intuitive and technically easy to work with. But I have only been able to do so since most categories I know are locally small.
If your category isn't locally small, then $f_*$ is in general not a function, but a functional. It is perfectly fine to define a functional $F$ to be injective iff it satisfies : $$\forall a, b, \ F(a) = F(b) \Longrightarrow a = b $$ in other words, $f_*$ is injective iff $$\forall a, b \in \operatorname{Hom}(x, c), \ f \circ a = f \circ b \Longrightarrow a = b $$ But that is the definition of $f$ being monic.