Kerodon

Exercise 7.6.3.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, so that Corollary 7.6.3.9 supplied an identification of $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}}^{\mathrm{h}} \operatorname{\mathcal{C}}$ with the fiber product of $\operatorname{\mathcal{C}}$ with itself over $\operatorname{\mathcal{D}}$ in the $\infty $-category $\operatorname{\mathcal{QC}}$. Show that, under this identification, the relative diagonal of $F$ (in the sense of Notation 7.6.2.15) is represented by the inclusion map $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}}^{\mathrm{h}} \operatorname{\mathcal{C}}$. Moreover, if $F$ is an isofibration, then we can replace the homotopy fiber product $\operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ with the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ (formed in the ordinary category of simplicial sets); see Corollary 4.5.2.27.