Subobject Class in Category of Sets

Theorem
Let $\mathbf {Set}$ be the category of sets.

Let $S$ be a set.

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

Let $\powerset S$ be the order category on the power set of $S$ induced by Subset Relation on Power Set is Partial Ordering.

Then $\map {\overline {\mathbf {Sub} }_{\mathbf {Set} } } S \cong \powerset S$.