Kerodon

Example 9.1.7.12. Let $\operatorname{\mathcal{C}}$ be a small category. For each object $C \in \operatorname{\mathcal{C}}$, let $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ denote the functor corepresented by $C$ (given on objects by the formula $h^ C(D) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$). If $A \hookrightarrow B$ is a monomorphism of simplicial sets, then the natural transformation $\underline{A} \times h^{C} \hookrightarrow \underline{B} \times h^{C}$ is a projective cofibration in $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$; here $\underline{A}$ and $\underline{B}$ denote the constant simplicial sets taking the values $A$ and $B$, respectively.