Definition 7.5.9.8. Let $\operatorname{\mathcal{C}}$ be a small category and let $\alpha : \mathscr {F}' \rightarrow \mathscr {F}$ be a natural transformation between diagrams $\mathscr {F}, \mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. We say that $\alpha $ is a projective cofibration if it is left semiorthgonal to all levelwise trivial Kan fibrations (see Remark 4.5.6.2). That is, $\alpha $ is a projective cofibration if every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \mathscr {F}' \ar [d]^{\alpha } \ar [r] & \mathscr {G}' \ar [d]^{ \beta } \\ \mathscr {F} \ar [r] \ar@ {-->}[ur] & \mathscr {G} } \]
admits a solution, under the assumption that $\beta $ is a levelwise trivial Kan fibration between diagrams $\mathscr {G}, \mathscr {G}': \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$.