Proposition 5.1.6.7. Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^-{q} & \operatorname{\mathcal{C}}' \ar [d]^-{q'} \\ \operatorname{\mathcal{D}}\ar [r]^-{\overline{F}} & \operatorname{\mathcal{D}}'. } \]
Assume that:
- $(1)$
The functors $q$ and $q'$ are inner fibrations.
- $(2)$
The inner fibration $q$ is a cartesian fibration and the functor $F$ carries $q$-cartesian morphisms of $\operatorname{\mathcal{C}}$ to locally $q'$-cartesian morphisms of $\operatorname{\mathcal{C}}'$.
- $(3)$
The functor $\overline{F}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}'$ is fully faithful.
Then $F$ is fully faithful if and only if, for every object $D \in \operatorname{\mathcal{D}}$ having image $D' = \overline{F}(D) \in \operatorname{\mathcal{D}}'$, the induced map of fibers $F_{D}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}'_{D'}$ is fully faithful.
Proof.
The “only if” direction follows from Proposition 4.6.2.8. For the converse, assume that each of the functors $F_{D}$ is fully faithful; we will show that $F$ is fully faithful. Let $X$ and $Z$ be objects of $\operatorname{\mathcal{C}}$ having images $\overline{X}, \overline{Z} \in \operatorname{\mathcal{D}}$; we wish to show that the upper horizontal map in the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( F(X), F(Z) ) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Z} ) \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}'}( \overline{F}( \overline{X} ), \overline{F}( \overline{Z} )) } \]
is a homotopy equivalence. Since $q$ and $q'$ are inner fibrations, the vertical maps are Kan fibrations (Proposition 4.6.1.21). Assumption $(3)$ guarantees that the lower horizontal map is a homotopy equivalence. By virtue of Proposition 3.2.8.1, it will suffice to show that for every morphism $\overline{e}: \overline{X} \rightarrow \overline{Z}$ in $\operatorname{\mathcal{D}}$, the induced map of fibers
\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\overline{e}} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( F(X), F(Z) )_{ \overline{F}(\overline{e} ) } \]
is a homotopy equivalence.
Let $[\theta ]$ denote the homotopy class of $\theta $, regarded as a morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. Since $q$ is a cartesian fibration, there exists a $q$-cartesian morphism $g: Y \rightarrow Z$ of $\operatorname{\mathcal{C}}$ satisfying $q(g) = \overline{e}$. We then have a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{ \overline{X} }}(X,Y) \ar [d]^-{ [g] \circ } \ar [r] & \operatorname{Hom}_{ \operatorname{\mathcal{C}}'_{ \overline{F}(\overline{X})} }( F(X), F(Y) ) \ar [d]^-{ [F(g)] \circ } \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{ \overline{e} } \ar [r]^-{ [ \theta ] } & \operatorname{Hom}_{\operatorname{\mathcal{C}}'}( F(X), F(Z) )_{ \overline{F}( \overline{e} ) } } \]
in $\mathrm{h} \mathit{\operatorname{Kan}}$, where the vertical maps are given by the composition law of Notation 5.1.3.10. Assumption $(2)$ guarantees that $F(g)$ is locally $q'$-cartesian, so that the vertical maps in this diagram are isomorphisms in $\mathrm{h} \mathit{\operatorname{Kan}}$ (Proposition 5.1.3.11). It will therefore suffice to show that the functor $F_{\overline{X}}$ induces a homotopy equivalence of mapping spaces $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{ \overline{X} }}(X,Y) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{C}}'_{ \overline{F}(\overline{X})} }( F(X), F(Y) )$, which follows from our assumption that $F_{ \overline{X} }$ is fully faithful.
$\square$