Category of Subobject Classes is Order 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 an order category.

Proof

This theorem requires a proof. In particular: Preliminary (rather, partial) results missing You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.