Construction 7.4.1.13. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, let $\operatorname{Fun}_{/ \operatorname{\mathcal{C}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ be the simplicial set parametrizing sections of $U$, and let
\[ \operatorname{ev}: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}\quad \quad (F, C) \mapsto F(C) \]
be the evaluation functor. Suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ and a natural transformation $\alpha : \underline{ \Delta ^{0} }_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F}|_{\operatorname{\mathcal{E}}}$. The image of $\alpha $ under the functor
\[ \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{S}}) \xrightarrow { \circ \operatorname{ev}} \operatorname{Fun}( \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \times \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \]
can be identified with a comparison morphism
\[ T: \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \underline{ \Delta ^0 }_{\operatorname{\mathcal{C}}}, \mathscr {F} ). \]