Category of Subobject Classes is Category

Theorem
Let $\mathbf C$ be a metacategory.

Let $C$ be an object of $\mathbf C$.

Let $\overline{\mathbf{Sub}}_{\mathbf C} \left({C}\right)$ be the category of subobject classes of $C$.

Then $\overline{\mathbf{Sub}}_{\mathbf C} \left({C}\right)$ is a metacategory.