\oset{\phi_L}{\lat} I \rat F$ and $\alpha$ is the resulting mediating arrow.

$$\xymatrix{
        &  L \ar[dl]_{\phi} \ar@{ >->}[d] & I \ar@{ >->}[d]
        \ar@{ >->}[l]_{\phi_L}\\
        A & {\mat{\phi}} \ar[l]_{\psi} & F
        \itul{\rm\footnotesize\color{gray}
          (FPBC)}
        \ar@{>->}[l] }
      \label{eq:fpbc-al-i}$$

$$\xymatrix{
        & L \ar[dl]_{\phi}   \ar@{ >->}[d] & L \ar@{ >->}[d]_{n_L}
        \ar@{ >->}[l]_{\mathit{id}_L} & I \ar@{ >->}[d]
        \ar@{ >->}[l]_{\phi_L} \\
        A & {\mat{\phi}} \ar[l]_{\psi} & {\rmat{\phi}{\phi_L}}
        \ar@{>.>}[l]_(.5){\alpha} 
        & F \ar@{ >->}[l]_(.4){\beta} \ar@/^1pc/@{ >->}[ll]
        \itull{\rm\footnotesize\color{gray} \hspace*{0.1cm}(PO)}
      }
      \label{eq:po-l-fpbc}$$

We come back to the running example (see Ex. \[ex:POCexample\]) and, as in Prop. \[prop:mat-rewritable\], determine the final pullback complement $I \rat F \rat {\mat{\phi}}$ of $I \oset{\phi_L}{\rat} L \rat \mat{\phi}$ (see diagram below left) and obtain $\rmat{\phi}{\phi_L}$ by taking the pushout over $L \lat I \rat F$ (see diagram below right).

It remains to be shown that $L\rat \rmat{\phi}{\phi_L}\to A$ represents every factorization which can be rewritten. As before we obtain a characterization of the rewritable objects, including the match, as the language of an arrow.

\[prop:language-rewriting-materialization\] [ Assume there is a production $p\colon L\oset{\phi_L}{\lat} I\oset{\phi_R}{\rat} R$ and let $L\oset{n_L}{\rat}\rmat{\phi}{\phi_L}$ be the match for the rewritable materialization for $\phi$ and $\phi_L$. Then we have $$\mathcal{L}(L\oset{n_L}{\rat} \rmat{\phi}{\phi_L}) =
    \{L\oset{m_L}{\rat} X\mid \exists \psi\colon (X\to A).\ (\phi =
    \psi \circ m_L \land 