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Proposition 7.1.4.19. 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 4.6.1.19). Applying Corollary 3.3.7.3, 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 4.6.1.19), 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 3.3.7.3). 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 5.1.2.1, 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. $\square$