Proof.
We will give a proof of $(2)$; assertion $(1)$ follows by a similar argument. We proceed as in the proof of Lemma 4.4.4.8. Suppose we are given an integer $n \geq 2$; we wish to show that every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{0} \ar [r]^-{ \sigma _0 } \ar [d] & \operatorname{Fun}(B,X) \ar [d]^-{q'} \\ \Delta ^ n \ar@ {-->}[ur]^{\sigma } \ar [r]^-{\overline{\sigma } } & \operatorname{Fun}(B,S) } \]
admits a solution, provided that the composite map
\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \hookrightarrow \Lambda ^ n_0 \xrightarrow {\sigma _0} \operatorname{Fun}(B,X) \]
is the edge $e$. Unwinding the definitions, we can rewrite this as a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ B \times \Lambda ^{n}_0 \ar [r]^-{F_0} \ar [d] & X \ar [d]^-{q} \\ B \times \Delta ^ n \ar [r]^-{ \overline{F} } \ar@ {-->}[ur]^{F} & S. } \]
Let $P$ denote the collection of all pairs $(A, F_ A)$, where $A \subseteq B$ is a simplicial subset and $F_ A: A \times \Delta ^ n \rightarrow X$ is a morphism of simplicial sets satisfying
\[ F_{A} |_{ A \times \Lambda ^{n}_0} = F_0 |_{ A \times \Lambda ^{n}_{0}} \quad \quad q \circ F_ A = \overline{F}|_{ A \times \Delta ^ n} \]
We regard $P$ as partially ordered set, where $(A, F_ A) \leq (A', F_{A'} )$ if $A \subseteq A'$ and $F_ A = F_{A'} |_{A \times \Delta ^ n}$. The partially ordered set $P$ satisfies the hypotheses of Zorn's lemma, and therefore has a maximal element $(A_{\mathrm{max}}, F_{A_{\mathrm{max}}})$. We will complete the proof by showing that $A_{\mathrm{max}} = B$. Assume otherwise. Then there exists some nondegenerate $m$-simplex $\tau : \Delta ^{m} \rightarrow B$ whose image is not contained in $A_{ \mathrm{max} }$. Choosing $m$ as small as possible, we can assume that $\tau $ carries the boundary $\operatorname{\partial \Delta }^{m}$ into $A_{\mathrm{max}}$. Let $A' \subseteq B$ be the union of $A_{\mathrm{max}}$ with the image of $\tau $, so that we have a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [d] \ar [r] & A_{\mathrm{max}} \ar [d] \\ \Delta ^{m} \ar [r] & A'. } \]
We will complete the proof by showing that the lifting problem
\[ \xymatrix@R =50pt@C=100pt{ (A_{\mathrm{max}} \times \Delta ^ n) {\coprod }_{ (A_{\mathrm{max}} \times \Lambda ^ n_0 )} (A' \times \Lambda ^ n_0) \ar [r]^-{(F_{A_{\mathrm{max}}}, F_0|_{A' \times \Lambda ^{n}_{0}})} \ar [d] & X \ar [d]^-{q} \\ A' \times \Delta ^ n \ar [r] \ar@ {-->}[ur] & S } \]
admits a solution (contradicting the maximality of the pair $(A_{\mathrm{max}}, F_{A_{\mathrm{max}}} )$).
Choose a sequence of simplicial subsets
\[ Y(0) \subset Y(1) \subset Y(2) \subset \cdots \subset Y(t) = \Delta ^{m} \times \Delta ^{n} \]
satisfying the requirements of Lemma 4.4.4.7, so that $F_{A_{\mathrm{max}}}$ determines a map of simplicial sets $G_0: Y(0) \rightarrow X$. We will show that, for $0 \leq s \leq t$, there exists a morphism of simplicial sets $G_{s}: Y(s) \rightarrow X$ satisfying $G_ s|_{Y(0)} = G_0$ and $q \circ G_ s = \overline{F}|_{ Y(s) }$ (in the case $s = t$, this will complete the proof of Lemma 5.2.1.5). We proceed by induction on $s$, the case $s=0$ being vacuous. Assume that $s > 0$ and that we have already constructed a morphism $G_{s-1}: Y(s-1) \rightarrow X$ satisfying $G_{s-1}|_{Y(0)} = F_0$ and $q \circ G_{s-1} = \overline{F}|_{ Y(s-1)}$. By construction, there exist integers $\ell \geq 2$, $0 \leq k< \ell $ and a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{\ell }_{k} \ar [r]^-{\tau _0} \ar [d] & Y(s-1) \ar [d] \\ \Delta ^{\ell } \ar [r]^-{\tau } & Y(s). } \]
Moreover, in the special case $k=0$, we can assume that $\tau (0) = (0,0)$ and $\tau (1) = (0,1)$, so that the composite map
\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \hookrightarrow \Lambda ^{\ell }_{k} \xrightarrow {\sigma _0} Y(s-1) \xrightarrow { G_{s-1} } X \]
corresponds to a $q$-cocartesian edge $e'$ of $X$. To construct the desired extension $F_ s$, it suffices to solve the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{\ell }_{k} \ar [r]^-{ G_{s-1} \circ \tau _0} \ar [d] & X \ar [d]^-{q} \\ \Delta ^{\ell } \ar [r]^-{ \overline{F} \circ \tau } \ar@ {-->}[ur] & S. } \]
For $0 < k < \ell $, the existence of the desired solution follows from our assumption that $q$ is an inner fibration; when $k = 0$, it follows from the fact that $e'$ is $q$-cocartesian.
$\square$