$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 4.3.6.15 (Joyal [MR1935979]). Let $p,q \geq 0$ be nonnegative integers. Then:
Assume $p > 0$. Then, for $0 \leq i \leq p$, the pushout-join monomorphism
\[ (\Lambda ^{p}_{i} \star \Delta ^ q) \coprod _{ ( \Lambda ^{p}_{i} \star \operatorname{\partial \Delta }^ q) } ( \Delta ^ p \star \operatorname{\partial \Delta }^ q) \hookrightarrow \Delta ^{p} \star \Delta ^ q \]
of Construction 4.3.6.3 is isomorphic to the horn inclusion $\Lambda ^{p+1+q}_{i} \hookrightarrow \Delta ^{p+1+q}$.
Assume $q> 0$. Then, for $0 \leq j \leq q$, the pushout-join monomorphism
\[ (\operatorname{\partial \Delta }^ p \star \Delta ^ q) \coprod _{ ( \operatorname{\partial \Delta }^ p \star \Lambda ^{q}_{j} ) } ( \Delta ^ p \star \Lambda ^ q_ j) \hookrightarrow \Delta ^{p} \star \Delta ^ q \]
of Construction 4.3.6.3 is isomorphic to the horn inclusion $\Lambda ^{p+1+q}_{p+1+j} \hookrightarrow \Delta ^{p+1+q}$.
Proof.
We will prove the first assertion; the second follows by symmetry. We begin by observing that there is a unique isomorphism of simplicial sets $u: \Delta ^{p} \star \Delta ^{q} \simeq \Delta ^{p+1+q}$ (Example 4.3.3.23). Let $\sigma $ be an $n$-simplex of the join $\Delta ^{p} \star \Delta ^{q}$; we wish to show that $u(\sigma )$ belongs to the horn $\Lambda ^{p+1+q}_{i}$ if and only if $\sigma $ belongs to the union of the simplicial subsets
\[ \Lambda ^{p}_{i} \star \Delta ^{q} \subseteq \Delta ^{p} \star \Delta ^{q} \supseteq \Delta ^{p} \star \operatorname{\partial \Delta }^{q}. \]
We consider three cases (see Remark 4.3.3.15):
The simplex $\sigma $ belongs to the simplicial subset $\Delta ^{p} \subseteq \Delta ^{p} \star \Delta ^{q}$. In this case, $\sigma $ is contained in $\Delta ^{p} \star \operatorname{\partial \Delta }^{q}$ and $u(\sigma )$ is contained in $\Lambda ^{p+1+q}_{i}$.
The simplex $\sigma $ belongs to the simplicial subset $\Delta ^{q} \subseteq \Delta ^{p} \star \Delta ^{q}$. In this case, $\sigma $ is contained in $\Lambda ^{p}_{i} \star \Delta ^{q}$ and $u(\sigma )$ is contained in $\Lambda ^{p+1+q}_{i}$ (since $p > 0$).
The simplex $\sigma $ factors as a composition
\[ \Delta ^{n} = \Delta ^{p' + 1 + q'} \simeq \Delta ^{p'} \star \Delta ^{q'} \xrightarrow { \sigma _{-} \star \sigma _{+} } \Delta ^{p} \star \Delta ^{q}. \]
Let us abuse notation by identifying $\sigma _{-}$ and $\sigma _{+}$ with nondecreasing functions $[p'] \rightarrow [p]$ and $[q'] \rightarrow [q]$, and $u(\sigma )$ with the nondecreasing function $[n] \rightarrow [p+1+q]$ given by their join. In this case, $\sigma $ fails to belong to the union $(\Lambda ^{p}_{i} \star \Delta ^{q}) \cup ( \Delta ^{p} \star \operatorname{\partial \Delta }^{q} )$ if and only if both of the following conditions are satisfied:
Together, these are equivalent to the assertion that the image of the nondecreasing function $u(\sigma ): [n] \rightarrow [p+1+q]$ contains $[p+1+q] \setminus \{ i\} $: that is, it fails to belong to the horn $\Lambda ^{p+1+q}_{i} \subseteq \Delta ^{p+1+q}$.
$\square$