Corollary 9.1.7.15. Let $\operatorname{\mathcal{C}}$ be a small category, and let $S$ be the collection of all projective cofibrations in the category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$. Then $S$ is the smallest weakly saturated collection of morphisms which contains each of the inclusion maps $\iota _{n,C}: \underline{\operatorname{\partial \Delta }^{n}} \times h^{C} \hookrightarrow \underline{\Delta ^{n}} \times h^{C}$, for each $n \geq 0$ and each object $C \in \operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. It follows from Remark 9.1.7.13 that $S$ is weakly saturated. Let $S'$ be the smallest weakly saturated collection of morphisms of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ which contains each $\iota _{n,C}$. Using Example 9.1.7.12, we see that $S'$ is contained in $S$. For every monomorphism of simplicial sets $A \hookrightarrow B$ and every object $C \in \operatorname{\mathcal{C}}$, Proposition 1.5.5.14 guarantees that the projective cofibration $\underline{A} \times h^{C} \hookrightarrow \underline{B} \times h^{C}$ is contained in $S'$. It follows from the proof of Proposition 9.1.7.14 that every morphism $\alpha _0: \mathscr {F}_0 \rightarrow \mathscr {G}$ in $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ factors as a composition $\mathscr {F}_0 \xrightarrow {\beta } \mathscr {F} \xrightarrow {\alpha } \mathscr {G}$, where $\beta $ belongs to $S'$ and $\alpha $ is a trivial Kan fibration. If $\alpha _0$ is projective cofibration, then the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \mathscr {F}_{0} \ar [d]^{ \alpha _0 } \ar [r]^-{\beta } & \mathscr {F} \ar [d]^{ \alpha } \\ \mathscr {G} \ar@ {-->}[ur] \ar@ {=}[r] & \mathscr {G} } \]
admits a solution. It follows that $\alpha _0$ is a retract of the morphism $\beta $, and therefore belongs to $S'$. $\square$