topos by taking final pullback complements.

\[cor:term-constr\]

Let $\varphi \colon L \to A$ be an arrow of $\C$ and let $\mathtt{true}_A \colon A \rightarrowtail A \times \Omega$ be the subobject classifier in the slice category $\C \downarrow A$ from $\textit{id}_A \colon A \to A$ to the projection $\pi_1 \colon A \times \Omega \to A$[ (see Fact \[prop:subobclass-slice\] in App. \[apx:topos\])]{}.

Then the terminal object $L \oset{\eta_\phi}{\rat} {\mat{\phi}} \oset{\psi}{\to} A$ in the materialization category consists of the arrows $\eta_\phi$ and $\psi = \pi_1\circ \chi_{\eta_\phi}$, where $L \oset{\eta_\phi}{\rightarrowtail} {\mat{\phi}}
    \oset{\chi_{\eta_\phi}}{\to} A \times \Omega$ is the final pullback complement of $L \oset{\varphi}{\to} A \oset{\mathtt{true}_A}{\rightarrowtail} A
    \times \Omega$.

$$\xymatrix{ L \ar@{->}[d]_{\phi} \ar@{ >.>}[r]^(.6){\eta_\phi
          \hspace*{.3cm}} & {\mat{\phi}}
        \ar@{.>}[d]^{\chi_{\eta_\phi}} \ar@{.>}[dr]^{\psi}&  \\
        A \ar@{ >.>}[r]_{\mathtt{true}_A \hspace*{.3cm}} & A \times
        \Omega \ar@{->}[r]_(.6){\pi_1}
        \itull{\mbox{\rm\footnotesize\color{gray}(FPBC)}} & A }$$

\[ex:construction-materialization\] We construct the materialization $L \oset{\eta_\phi}{\rat} {\mat{\phi}}
    \oset{\psi}{\to} A$ for the following morphism\

$\varphi \colon L \to A$ of graphs with a single (omitted) label:

In particular, the materialization is obtained as a final pullback complement as depicted to the right (compare with the corresponding diagram in Corollary \[cor:term-constr\]). Note

that edges which are not in the image of $\eta_\phi$ resp. $\mathtt{true}_A$ are dashed.

This construction corresponds to the usual intuition 