Example 7.4.1.10. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a diagram. Then the coslice diagonal
\[ \int _{\operatorname{\mathcal{C}}} \mathscr {F} = \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}_{\Delta ^0 / } \hookrightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{S}}} ( \{ \Delta ^0 \} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}) = \{ \Delta ^0 \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{C}} \]
can be identified with a natural transformation
\[ \alpha : \underline{ \Delta ^0 }_{\int _{\operatorname{\mathcal{C}}} \mathscr {F} } \rightarrow \mathscr {F}|_{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} } \]
which exhibits $\mathscr {F}$ as a covariant transport representation for the left fibration $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$.