\begin{definition}[Definition:Local Membership Relation]
Let $\mathbf C$ be a metacategory.
Let $C$ be an object of $\mathbf C$.
The '''local membership relation''' $\in_C$ between variable elements $z: Z \to C$ and subobjects $m: M \to C$ of $C$ is defined by:
:$z \in_C m$ {{iff}} there exists an $f: Z \to M$: $z = m \circ f$.
If $z \in_C m$, one says that $z$ is a '''local member''' of $m$.
\end{definition}