Proof.
We first show that $(1)$ implies $(2)$. Assume that $(1)$ is satisfied and let $B \rightarrow \operatorname{\mathcal{E}}$ be a morphism of simplicial sets; we wish to show that the induced map $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{D}})$ is an equivalence of $\infty $-categories. By virtue of Theorem 4.5.7.1, it will suffice to show that for every simplicial set $B'$, the induced map
\[ \operatorname{Fun}( B', \operatorname{Fun}_{/\operatorname{\mathcal{E}}}( B, \operatorname{\mathcal{C}}) )^{\simeq } \rightarrow \operatorname{Fun}( B', \operatorname{Fun}_{/\operatorname{\mathcal{E}}}( B, \operatorname{\mathcal{D}}) )^{\simeq } \]
is a homotopy equivalence of Kan complexes. This follows by applying $(1)$ to the composite map $B' \times B \rightarrow B \rightarrow \operatorname{\mathcal{E}}$.
We now prove that $(2)$ implies $(3)$. Assume that condition $(2)$ is satisfied. Setting $B=\operatorname{\mathcal{D}}$, we deduce that composition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{D}},\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{D}},\operatorname{\mathcal{D}})$. In particular, there exists a morphism $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ in $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{E}}}$ such that $F \circ G$ is isomorphic to $\operatorname{id}_{\operatorname{\mathcal{D}}}$ as an object of the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{D}},\operatorname{\mathcal{D}})$. It follows that $F \circ G \circ F$ is isomorphic to $F$ as an object of the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}},\operatorname{\mathcal{D}})$. Applying condition $(2)$ in the case $B = \operatorname{\mathcal{C}}$, we see that postcomposition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}},\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}},\operatorname{\mathcal{D}})$, so that $G \circ F$ is isomorphic to $\operatorname{id}_{\operatorname{\mathcal{C}}}$ as an object of $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$. It follows that $G$ is a homotopy inverse of $F$ relative to $\operatorname{\mathcal{E}}$. In particular, $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$.
The implication $(3) \Rightarrow (4)$ follows by combining Remark 5.1.7.4 with Corollary 5.1.7.8. We will complete the proof by showing that $(4)$ implies $(1)$. Assume that condition $(4)$ is satisfied, and let $B$ be a simplicial set equipped with a morphism $B \rightarrow \operatorname{\mathcal{E}}$. We wish to show that composition with $F$ induces a homotopy equivalence of Kan complexes $\theta _{B}: \operatorname{Fun}_{/\operatorname{\mathcal{E}}}( B, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{D}})^{\simeq }$. Assume first that the simplicial set $B$ has dimension $\leq n$, for some integer $n \geq -1$. Our proof proceeds by induction on $n$. If $n=-1$, then $B$ is empty and there is nothing to prove. We may therefore assume without loss of generality that $n \geq 0$. Let $A$ be the $(n-1)$-skeleton of $B$. Our inductive hypothesis guarantees that $\theta _{A}$ is a homotopy equivalence. By virtue of Proposition 3.2.8.1, it will suffice to verify the following:
- $(\ast )$
The restriction maps
\[ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{C}}) \quad \quad \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{D}}) \]
are isofibrations of $\infty $-categories, and therefore induce Kan fibrations
\[ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{C}})^{\simeq } \quad \quad \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{D}})^{\simeq } \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{D}})^{\simeq }; \]
see Proposition 4.4.3.7.
- $(\ast ')$
For every object $T \in \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{C}})$, the induced map of fibers
\[ \{ T\} \times _{ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{C}})} \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{C}}) \rightarrow \{ F \circ T\} \times _{ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{D}})} \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{D}}) \]
is an equivalence of $\infty $-categories, and therefore induces a homotopy equivalence of Kan complexes
\[ \{ T\} \times _{ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{C}})^{\simeq }} \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{C}})^{\simeq } \rightarrow \{ F \circ T\} \times _{ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{D}})^{\simeq }} \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{D}})^{\simeq } \]
(see Remark 4.5.1.19).
Let $J$ denote the set of all nondegenerate $n$-simplices of $B$. Proposition 1.1.4.12 supplies a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \coprod _{\sigma \in J} \operatorname{\partial \Delta }^ n \ar [r] \ar [d] & \coprod _{\sigma \in J} \Delta ^ n \ar [d] \\ A \ar [r] & B. } \]
Consequently, to verify $(\ast )$ and $(\ast ')$, we can assume without loss of generality that $B = \Delta ^ n$ is a standard simplex and that $A = \operatorname{\partial \Delta }^ n$ is its boundary. Replacing $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ by the fiber products $\Delta ^ n \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{C}}$ and $\Delta ^ n \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$, we can reduce further to the case where $\operatorname{\mathcal{E}}= \Delta ^ n$ is a standard simplex. Applying Example 4.4.1.6, we deduce that $U$ and $V$ are isofibrations, so that assertion $(\ast )$ follows from Proposition 4.4.5.1. Invoking assumption $(4)$, we deduce that $F$ is an equivalence of $\infty $-categories, and therefore induces equivalences
\[ \operatorname{Fun}(A, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{D}}) \quad \quad \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{D}}). \]
Assertion $(\ast ')$ now follows from Corollary 4.5.2.32.
We now treat the case where $B$ is a general simplicial set. For each $n \geq 0$, let $\operatorname{sk}_{n}(B)$ denote the $n$-skeleton of $B$ (Construction 1.1.4.1). Using $(\ast )$ and Corollary 4.5.6.22, we see that $\theta _{B}$ can be realized as the inverse limit of a tower
\[ \xymatrix@R =50pt@C=25pt{ \cdots \ar [r] & \operatorname{Fun}_{/S}( \operatorname{sk}_2(B), X)^{\simeq } \ar [d]^{ \theta _{\operatorname{sk}_2(B)}} \ar [r] & \operatorname{Fun}_{/S}( \operatorname{sk}_1(B), X)^{\simeq } \ar [d]^{ \theta _{\operatorname{sk}_1(B)} } \ar [r] & \operatorname{Fun}_{/S}( \operatorname{sk}_0(B), X)^{\simeq } \ar [d]^{ \theta _{\operatorname{sk}_0(B)} } \\ \cdots \ar [r] & \operatorname{Fun}_{/S}( \operatorname{sk}_2(B), X')^{\simeq } \ar [r] & \operatorname{Fun}_{/S}( \operatorname{sk}_1(B), X')^{\simeq } \ar [r] & \operatorname{Fun}_{/S}( \operatorname{sk}_0(B), X')^{\simeq }, } \]
where each of the transition maps is a Kan fibration. The preceding arguments show that each of the vertical maps $\theta _{ \operatorname{sk}_ n(B) }$ is a homotopy equivalence of Kan complexes. Invoking Example 4.5.6.18, we deduce that $\theta _{B}$ is a homotopy equivalence of Kan complexes.
$\square$