Category of Subobject Classes is Order Category

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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 an order category.


Proof