into account, we consider a subcategory of the materialization category.

\[def:materialization-category-rewritable\] Let $\phi\colon L\to A$ be an arrow of $\C$ and let $\phi_L \colon I\rat L$ be a mono (corresponding to the left leg of a production). The *materialization subcategory of rewritable objects* for $\phi$ and $\phi_L$, denoted $\matcat[\phi_L]{\phi}$, is the full subcategory of $\matcat{\phi}$ containing as objects all factorizations $L\oset{m}{\rat} X\to A$ of $\phi$, where $m$ is a mono and $I\oset{\phi_L}{\rat} L\oset{m}{\rat} X$ has a pushout complement.

Its terminal element, if it exists, is denoted by $L\oset{n_L}{\rat}\rmat{\phi}{\phi_L} \to A$ and is called the *rewritable materialization*.

We will show that this subcategory of the materialization category has a terminal object.

\[prop:mat-rewritable\]

Let $\phi\colon L\to A$ be an arrow and let $\phi_L\colon I\rat L$ be a mono of $\C$. Then the *rewritable materialization of $\phi$ w.r.t. $\phi_L$* exists and can be constructed as the following factorization $L\oset{n_L}{\rat} \rmat{\phi}{\phi_L} \oset{\psi\circ
    \alpha}{\longrightarrow} A$ of $\phi$. In the left diagram, $F$ is obtained as the final pullback complement of $I \oset{\phi_L}{\rat} L \rat {\mat{\phi}}$, where $L \rat {\mat{\phi}} \oset{\psi}{\to} A$ is the materialization of $\phi$ (Def. \[def:materialization-category\]). Next in the right diagram $L \oset{n_L}{\rat} \rmat{\phi}{\phi_L} \oset{\beta}{\lat} F$ is the pushout of the span $L 