$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 3.3.5.9. Let $J$ be a nonempty finite set, let $P(J)$ denote the collection of subsets of $J$ (partially ordered by inclusion), and set $P_{-}(J) = P(J) \setminus \{ J \} $. Then the inclusion of simplicial sets
\[ \theta : \operatorname{N}_{\bullet }( P_{-}(J) ) \hookrightarrow \operatorname{N}_{\bullet }( P(J) ) = \operatorname{\raise {0.1ex}{\square }}^{J} \]
is anodyne.
Proof.
Fix an element $j \in J$ and set $I = J \setminus \{ j\} $, so that the simplicial cube $\operatorname{\raise {0.1ex}{\square }}^ J$ can be identified with the product $\Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I} \simeq \Delta ^1 \times \operatorname{N}_{\bullet }( P(I) )$. Under this identification, $\theta $ corresponds to the inclusion map
\[ (\Delta ^1 \times \operatorname{N}_{\bullet }( P_{-}(I) )) \coprod _{ \{ 0\} \times \operatorname{N}_{\bullet }( P_{-}(I) )} ( \{ 0\} \times \operatorname{N}_{\bullet }( P(I) ) ) \hookrightarrow \Delta ^1 \times \operatorname{N}_{\bullet }( P(I) ), \]
which is anodyne by virtue of Proposition 3.1.2.9.
$\square$