Kerodon

Remark 4.6.1.14 (Morphism Spaces in Homotopy Fiber Products). Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. Let $X_0, Y_0 \in \operatorname{\mathcal{C}}_0$ and $X_1, Y_1 \in \operatorname{\mathcal{C}}_1$ be objects having the same images $F_0(X_0) = X = F_1(X_1)$ and $F_0(Y_0) = Y = F_1(Y_1)$ in $\operatorname{\mathcal{C}}$, so that $X_{01} = (X_0, X_1, \operatorname{id}_{X} )$ and $Y_{01} = (Y_0, Y_1, \operatorname{id}_{Y} )$ can be viewed as objects of the homotopy fiber product $\operatorname{\mathcal{C}}_{01} = \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{1}$ (see Construction 4.5.2.1). Then the mapping space $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{01} }( X_{01}, Y_{01} )$ can be identified with the homotopy fiber product of Kan complexes