Proof.
Unwinding the definitions, we can identify the diagram (5.37) with a morphism of simplicial sets
\[ \overline{f}: ( \Lambda ^{m}_{i} \star K ) {\coprod }_{ (\Lambda ^{m}_{i} \star K_0 ) } ( \Delta ^{m} \star K_0 ) \rightarrow \operatorname{\mathcal{C}}, \]
and we wish to show that $\overline{f}$ can be extended to a morphism $\Delta ^{m} \star K \rightarrow \operatorname{\mathcal{C}}$. Let $P$ be the collection of all pairs $(L,g)$, where $L$ is a simplicial subset of $K$ containing $K_0$ and $g: \Delta ^{m} \star L \rightarrow \operatorname{\mathcal{C}}$ is a morphism satisfying
\[ g|_{ \Delta ^{m} \star K_0} = \overline{f} |_{ \Delta ^{m} \star K_0} \quad \quad g|_{ \Lambda ^{m}_{i} \star L} = \overline{f} |_{ \Lambda ^{m}_{i} \star L}. \]
We regard $P$ as a partially ordered set, with $(L, g) \leq (L', g')$ if $L$ is contained in $L'$ and $g = g'|_{\Delta ^ m \star L}$. The partially ordered set $P$ satisfies the hypotheses of Zorn's lemma and therefore admits a maximal element $(L_{\mathrm{max}}, g_{\mathrm{max}} )$. We will complete the proof by showing that $L_{\mathrm{max}} = K$ (so that $g_{\mathrm{max}}$ is the desired extension of $\overline{f}$). Suppose otherwise. Then there is some nondegenerate simplex $\rho : \Delta ^{n} \rightarrow K$ which is not contained in $L_{\mathrm{max}}$. Choosing $\rho $ so that $n$ is as small as possible, we may assume without loss of generality that $\rho $ carries the boundary $\operatorname{\partial \Delta }^ n$ into $L_{\mathrm{max}}$. Let $L' \subseteq K$ be the simplicial subset given by the union of $L_{\mathrm{max}}$ together with the image of $\rho $, so that $\rho $ determines a pushout diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & L_{\mathrm{max}} \ar [d] \\ \Delta ^{n} \ar [r] & L'. } \]
We will show that $g_{\mathrm{max}}$ can be extended to a morphism of simplicial sets $g': \Delta ^{m} \star L' \rightarrow \operatorname{\mathcal{C}}$ satisfying $g'|_{ \Lambda ^{m}_{i} \star L'} = \overline{f}|_{ \Lambda ^{m}_{i} \star L'}$; thereby contradicting the maximality of $(L_{\mathrm{max}}, g_{\mathrm{max}} )$ and completing the proof of Proposition 5.4.3.8. Note that the composite maps
\[ \Lambda ^{m}_{i} \star \Delta ^{n} \xrightarrow { \operatorname{id}\star \rho } \Lambda ^{m}_{i} \star K \xrightarrow { \overline{f} } \operatorname{\mathcal{C}} \]
\[ \Delta ^{m} \star \operatorname{\partial \Delta }^ n \xrightarrow { \operatorname{id}\star \rho } \Delta ^{m} \star L_{\mathrm{max}} \xrightarrow { g_{\mathrm{max}}} \operatorname{\mathcal{C}} \]
can be amalgamated to a morphism of simplicial sets
\[ \tau _0: (\Lambda ^{m}_{i} \star \Delta ^{n}) {\coprod }_{ (\Lambda ^{m}_{i} \star \operatorname{\partial \Delta }^ n)} (\Delta ^ m \star \operatorname{\partial \Delta }^ n) \rightarrow \operatorname{\mathcal{C}}, \]
whose source can be identified with the horn $\Lambda ^{m+1+n}_{i} \subseteq \Delta ^{m+1+n}$ (Lemma 4.3.6.15). We wish to show that $\tau _0$ can be extended to a map
\[ \tau : \Delta ^{m} \star \Delta ^{n} \simeq \Delta ^{m+1+n} \rightarrow \operatorname{\mathcal{C}}. \]
If $0 < i \leq m$, the desired extension exists because the composite map
\[ \Delta ^{2} \simeq \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) \subseteq \Lambda ^{m+1+n}_{i} \xrightarrow {\tau _0} \operatorname{\mathcal{C}} \]
is a thin $2$-simplex of $\operatorname{\mathcal{C}}$ (by virtue of assumption $(b)$ when $i < m$ or $(c)$ in the case $i = m$). If $i=0$, then the desired extension exists because assumption $(a)$ guarantees that $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category and the $2$-simplex
\[ \Delta ^{2} \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 < m+1+n\} ) \subseteq \Lambda ^{m+1+n}_{i} \xrightarrow {\tau _0} \operatorname{\mathcal{C}} \]
is left-degenerate.
$\square$