# Category of Subobject Classes is Category

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## 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.

## Proof

This needs considerable tedious hard slog to complete it.A feast of checking well-definednessTo 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 `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |