is denoted by $L\rat {\mat{\phi}} \to A$ and is called the *materialization* of $\phi$.

Sometimes we will also call the object $\mat{\phi}$ the materialization of $\phi$, omitting the arrows.

Since we are working in a topos by assumption, the slice category over $A$ provides us with a convenient setting to construct materializations. Note in particular that in the diagram in Def. \[def:materialization-category\] above, the span $X\leftarrowtail L\rat L$ is a partial map from $X$ to $L$ in the slice category over $A$. Hence the materialization $\mat{\phi}$ corresponds to the partial map classifier for $L$ in this slice category.

\[prop:term-pmc\] [ Let $\phi\colon L \to~A$ be an arrow in $\C$, and let $\eta_\phi \colon \phi \to F(\phi)$, with $F(\phi) \colon \bar{A} \to A$, be the partial map classifier of $\phi$ in the slice category $\C\!\downarrow\! A$ (which also is a topos).[^3] Then $L \oset{\eta_\phi}{\to} \bar{A} \oset{F(\phi)}{\to} A$ is the materialization of $\phi$, hence $\mat{\phi} = \bar{A}$. ]{}

As a direct consequence of Prop. \[prop:fpbc-soc\] and Prop. \[prop:term-pmc\] (and the fact that final pullback complements in the slice category correspond to those in the base category [@l:graph-rewritinginspancat]), the terminal object of the materialization category can be constructed for each arrow of a 