Subobject Class in Category of Sets

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

Let $S$ be a set.

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

Let $\mathcal P \left({S}\right)$ be the poset category on the power set of $S$ induced by Subset Relation on Power Set is Partial Ordering.

Then $\overline{\mathbf{Sub}}_{\mathbf{Set}} \left({S}\right) \cong \mathcal P \left({S}\right)$.