\begin{definition}[Definition:Category of Subobjects]
Let $\mathbf C$ be a metacategory.
Let $C$ be an object of $\mathbf C$.
The '''category of subobjects of $C$''', denoted $\mathbf{Sub}_{\mathbf C} \left({C}\right)$, is defined as follows:
{{DefineCategory
|ob = Subobjects $m: B \to C$ of $C$
|mor = Morphisms $f: \operatorname{dom} m \to \operatorname{dom} m'$ of $\mathbf C$ such that $m' \circ f = m$ in $\mathbf C$
|comp = Inherited from $\mathbf C$
|id = $\operatorname{id}_m := \operatorname{id}_{\operatorname{dom} m}$, the identity morphism in $\mathbf C$ of the domain of $m$
}}
The behaviour of the morphisms is shown in the following commutative diagram in $\mathbf C$:
::$\begin{xy}\xymatrix@+1em{
 B
  \ar[r]^*+{f}
  \ar[rd]_*+{m}
&
 B'
  \ar[d]^*+{m'}
\\
&
 C
}\end{xy}$
\end{definition}