\begin{definition}[Definition:Category of Subobject Classes]
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:
{{DefineCategory
|ob = Subobject classes $\left[\!\left[{m}\right]\!\right]$ of $C$
|mor = Morphism classes $\left[\!\left[{f}\right]\!\right]: \left[\!\left[{m}\right]\!\right] \to \left[\!\left[{m'}\right]\!\right]$
|comp = Inherited from $\mathbf{Sub}_{\mathbf C} \left({C}\right)$, i.e. $\left[\!\left[{g}\right]\!\right] \circ \left[\!\left[{f}\right]\!\right] := \left[\!\left[{g \circ f}\right]\!\right]$
|id = $\operatorname{id}_{\left[\!\left[{m}\right]\!\right]} := \left[\!\left[{\operatorname{id}_m}\right]\!\right]$, the morphism class of the identity morphism of $m$ in $\mathbf{Sub}_{\mathbf C} \left({C}\right)$
}}
\end{definition}