Category of Subobject Classes is Category

Theorem
Let $\mathbf C$ be a metacategory.

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

Let $\map {\overline {\mathbf {Sub} }_{\mathbf C} } C$ be the category of subobject classes of $C$.

Then $\map {\overline {\mathbf {Sub} }_{\mathbf C} } C$ is a metacategory.