Proposition 7.1.4.16 (02WR)

Proposition 7.1.4.16. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:

$(1)$: An object $Y \in \operatorname{\mathcal{C}}$ is $U$-initial if and only if $U$ induces an equivalence of $\infty $-categories $U': \operatorname{\mathcal{C}}_{Y/} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ U(Y) / }$.
$(2)$: An object $Y \in \operatorname{\mathcal{C}}$ is $U$-final if and only if $U$ induces an equivalence of $\infty $-categories $U'': \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/U(Y)}$.

Proof. We will prove $(2)$; the proof of $(1)$ is similar. Fix an object $Y \in \operatorname{\mathcal{C}}$, so that the morphism $U''$ of $(1)$ fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{/Y} \ar [rr]^{U''} \ar [dr] & & \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / U(Y) } \ar [dl] \\ & \operatorname{\mathcal{C}}& } \]

where the vertical maps are right fibrations (Proposition 4.3.6.1). Applying Corollary 5.1.7.15, we see that $U''$ is an equivalence of $\infty $-categories if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the induced map of fibers

\[ U''_{X}: \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y} \rightarrow \{ X\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / U(Y) } \]

is a homotopy equivalence of Kan complexes. By virtue of Proposition 4.6.5.10, this is equivalent to the requirement that $U$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(X), U(Y) )$. $\square$