Kerodon

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Notation 7.3.1.1. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\delta : K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. For each object $C \in \operatorname{\mathcal{C}}$, we let $K_{/C}$ denote the fiber product $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$. Note that the slice diagonal of Construction 4.6.4.13 determines a map $K_{/C} \rightarrow K \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ C\}$, which we can identify with a natural transformation of diagrams $\gamma : \delta |_{ K_{/C} } \rightarrow \underline{C}$; here $\delta |_{K_{/C} }$ denotes the composition $K_{/C} \rightarrow K \xrightarrow {\delta } \operatorname{\mathcal{C}}$, while $\underline{C}$ denotes the constant diagram $K_{/C} \rightarrow \operatorname{\mathcal{C}}$ taking the value $C$. Similarly, we let $K_{C/}$ denote the fiber product $\operatorname{\mathcal{C}}_{C/} \times _{\operatorname{\mathcal{C}}} K$, so that the coslice diagonal of Construction 4.6.4.13 determines a natural transformation $\gamma ': \underline{C} \rightarrow \delta |_{ K_{C/} }$.