Definition:Category of Subobject Classes

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Let $\mathbf C$ be a metacategory.

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

Let $\map {\mathbf{Sub}_{\mathbf C} } C$ be the category of subobjects of $C$.

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

Objects:         Subobject classes $\eqclass m {}$ of $C$
Morphisms: Morphism classes $\eqclass f {}: \eqclass m {} \to \eqclass {m'} {}$
Composition: Inherited from $\map {\mathbf{Sub}_{\mathbf C} } C$: $\eqclass g {} \circ \eqclass f {} := \eqclass {g \circ f} {}$
Identity morphisms: $\operatorname{id}_{\eqclass m {} } := \eqclass {\operatorname{id}_m} {}$, the morphism class of the identity morphism of $m$ in $\map {\mathbf{Sub}_{\mathbf C} } C$

Also denoted as

Most authors don't care to distinguish the category of subobject classes symbolically from the category of subobjects $\map {\mathbf{Sub}_{\mathbf C} } C$.

Also see