Let $C'$ be the category with objects $C$ and morphism the monic morphisms of $C$. In any topos, $\phi_A: \operatorname{Sub}(A) \cong \operatorname{Hom}_C(A,\Omega)$ and $\phi_A(m) = \chi_m$. This seems to also be a natural transformation since if $m \text{ monic} \in \operatorname{Sub}(A)$ and $n : A \mapsto B$ is monic, then $n\circ m \in \operatorname{Sub}(B)$ and $\chi_{n \circ m} \circ n = \chi_m$ Therefore, $\phi$ behaves like a contravariant functor since the $\operatorname{Sub}$ maps arrows by composing similar to $\operatorname{Hom}$ (only contravariantly).
Is this important/ does it get used elsewhere?