Definition:Subobject Class/Morphism Class

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 $C$.

Define the equivalence $\sim$ on the morphisms of $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ as follows.

For morphisms $f: m \to n$ and $g: m' \to n'$ of $\mathbf{Sub}_{\mathbf C} \left({C}\right)$:


 * $f \sim g$ iff $m \sim m'$ and $n \sim n'$

where $m \sim m'$ signifies equivalence of subobjects.

That $\sim$ in fact is an equivalence is shown on Morphism Class Equivalence is Equivalence.

A morphism class is an equivalence class $\left[\!\left[{f}\right]\!\right]$ under $\sim$ of a morphism $f: m \to m'$.

The domain and codomain of $\left[\!\left[{f}\right]\!\right]$ are taken to be $\left[\!\left[{m}\right]\!\right]$ and $\left[\!\left[{m'}\right]\!\right]$, respectively.

Also see

 * Category of Subobject Classes, of which this are the morphisms.