Notation 7.7.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, which we identify with an object of the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. By definition, objects of the slice $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F}$ can be identified with pairs $(F', \gamma )$, where $F': K \rightarrow \operatorname{\mathcal{C}}$ is a diagram and $\gamma : F' \rightarrow F$ is a natural transformation.We let $\operatorname{Fun}(K,\operatorname{\mathcal{C}})_{/F}^{\operatorname{Cart}}$ denote the full subcategory of $\operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F}$ spanned by those pairs $(F',\gamma )$ where $\gamma $ is cartesian, in the sense of Definition 7.7.1.1.
Let $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be the functor given by evaluation at $1 \in \Delta ^1$. Then the slice diagonal morphism of Construction 4.6.4.13) induces an equivalence of $\infty $-categories
(see Corollary 4.6.4.18). If $\operatorname{\mathcal{C}}$ admits pullbacks, then $\operatorname{ev}_{1}$ is a cartesian fibration, and $\iota $ restricts to an equivalence of full subcategories $\operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F}^{\operatorname{Cart}} \hookrightarrow \operatorname{Fun}^{\operatorname{ACart}}_{/\operatorname{\mathcal{C}}}(K, \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ (see Remark 7.7.1.12).