Definition:Category of Subobject Classes

Definition
Let $\mathbf C$ be a metacategory.

Let $C$ be an object of $\mathbf C$.

Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$.

The category of subobject classes of $C$, denoted $\overline{\mathbf{Sub}}_{\mathbf C} \left({C}\right)$, is defined as follows:

Also denoted as
Most authors don't care to distinguish the category of subobject classes symbolically from the category of subobjects $\mathbf{Sub}_{\mathbf C} \left({C}\right)$.

Also see

 * Category of Subobject Classes is Category
 * Category of Subobjects