Proposition 7.1.5.19. Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [rr]^{F} \ar [dr]_{U} & & \operatorname{\mathcal{D}}\ar [dl]^{V} \\ & \operatorname{\mathcal{E}}, & } \]
where $U$ and $V$ are inner fibrations. Let $E \in \operatorname{\mathcal{E}}$ be an object, and let $F_{E}: \operatorname{\mathcal{C}}_{E} \rightarrow \operatorname{\mathcal{D}}_{E}$ denote the corresponding restriction of $F$. Then:
- $(1)$
If $X \in \operatorname{\mathcal{C}}_{E}$ is $F$-initial when viewed as an object of the $\infty $-category $\operatorname{\mathcal{C}}$, then $X$ is $F_{E}$-initial.
- $(2)$
Assume that $U$ and $V$ are cartesian fibrations, and that the functor $F$ carries $U$-cartesian morphisms of $\operatorname{\mathcal{C}}$ to $V$-cartesian morphisms of $\operatorname{\mathcal{D}}$. If $X$ is $F_{E}$-initial, then it is $F$-initial when viewed as an object of $\operatorname{\mathcal{C}}$.
Proof.
We first prove $(1)$. Assume that $X$ is $F$-initial. For every object $Y \in \operatorname{\mathcal{C}}_{E}$, we have a commutative diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \ar [rr]^{\rho } \ar [dr] & & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \ar [dl] \\ & \operatorname{Hom}_{\operatorname{\mathcal{E}}}( E, E ). & } \]
Our assumption that $X$ is $F$-initial guarantees that $\rho $ is a homotopy equivalence. Since $U$ and $V$ are inner fibrations, the vertical maps are Kan fibrations (Proposition 4.6.1.21). Applying Corollary 3.3.7.5, we conclude that $\rho $ restricts to a homotopy equivalence
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}_{E}}(X,Y) & = & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(E,E) } \{ \operatorname{id}_{E} \} \\ & \rightarrow & \operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(X),F(Y)) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(E,E) } \{ \operatorname{id}_{E} \} \\ & = & \operatorname{Hom}_{\operatorname{\mathcal{D}}_{E}}( F(X), F(Y) ). \end{eqnarray*}
Allowing $Y$ to vary over objects of $\operatorname{\mathcal{C}}_{E}$, it follows that $X$ is an $F_{E}$-initial object of $\operatorname{\mathcal{C}}$.
We now prove $(2)$. Assume that $U$ and $V$ are cartesian fibrations, that the functor $F$ carries $U$-cartesian morphisms of $\operatorname{\mathcal{C}}$ to $V$-cartesian morphisms of $\operatorname{\mathcal{D}}$, and that $X$ is $F_{E}$-initial. We wish to show that $X$ is $F$-initial. Fix an object $Z \in \operatorname{\mathcal{C}}$; we must show that the horizontal map in the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Z ) \ar [rr]^{\theta } \ar [dr] & & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Z) ) \ar [dl] \\ & \operatorname{Hom}_{\operatorname{\mathcal{E}}}( U(X), U(Z) ) & } \]
is a homotopy equivalence. Since the vertical maps are Kan fibrations (Proposition 4.6.1.21), it will suffice to show that the induced map
\[ \theta _{ \overline{f}}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( U(X), U(Z) ) } \{ \overline{f} \} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Z) ) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( U(X), U(Z) ) } \{ \overline{f} \} \]
is a homotopy equivalence, for each morphism $\overline{f}: U(X) \rightarrow U(Z)$ in the $\infty $-category $\operatorname{\mathcal{E}}$ (Corollary 3.3.7.5). Since $U$ is a cartesian fibration, we can write $\overline{f} = U(f)$, where $f: Y \rightarrow Z$ is a $U$-cartesian morphism in $\operatorname{\mathcal{C}}$. By assumption, the image $F(f): F(Y) \rightarrow F(Z)$ is a $V$-cartesian morphism in the $\infty $-category $\operatorname{\mathcal{D}}$. Using Proposition 5.1.3.11, we can replace $\theta _{ \overline{f} }$ with the morphism
\[ \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{E} }( X, Y) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{D}}_{E} }( F(X), F(Y) ), \]
which is a homotopy equivalence by virtue of our assumption that $X$ is $F_{E}$-initial.
$\square$