Kerodon

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Construction 4.6.5.6. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$, and let

\[ \delta _{X/}: \operatorname{\mathcal{C}}_{X/} \hookrightarrow \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\quad \quad \delta _{/Y}: \operatorname{\mathcal{C}}_{/Y} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} \]

be the coslice and slice diagonal morphisms of Construction 4.6.4.13. Restricting to the fibers over the objects $Y,X \in \operatorname{\mathcal{C}}$, we obtain morphisms of Kan complexes

\[ \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y) = \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]
\[ \operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y) = \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y} \rightarrow \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), \]

which we will denote by $\iota ^{\mathrm{L}}_{X,Y}$ and $\iota ^{\mathrm{R}}_{X,Y}$, respectively. We will refer to $\iota ^{\mathrm{L}}_{X,Y}$ as the left-pinch inclusion map and to $\iota ^{\mathrm{R}}_{X,Y}$ as the right-pinch inclusion map.