Proposition 7.5.9.12. Let $\operatorname{\mathcal{C}}$ be a small category and let $\alpha _0: \mathscr {F}_0 \rightarrow \mathscr {G}$ be a natural transformation between diagrams $\mathscr {F}_0, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Then $\alpha _0$ factors as a composition
\[ \mathscr {F}_0 \xrightarrow {\beta } \mathscr {F} \xrightarrow {\alpha } \mathscr {G}, \]
where $\beta $ is a projective cofibration and $\alpha $ is a levelwise trivial Kan fibration.
Proof.
We will construct $\mathscr {F}$ as the colimit of a diagram of projective cofibrations
\[ \mathscr {F}_{0} \rightarrow \mathscr {F}_{1} \rightarrow \mathscr {F}_{2} \rightarrow \mathscr {F}_{3} \rightarrow \cdots \]
in the category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})_{ / \mathscr {G} }$. Fix $n \geq 0$, and suppose that we have constructed an object $\mathscr {F}_{n} \in \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})_{ / \mathscr {G} }$, which we identify with a natural transformation $\alpha _{n}: \mathscr {F}_{n} \rightarrow \mathscr {G}$. For each object $C \in \operatorname{\mathcal{C}}$, Exercise 3.1.7.11 guarantees that $\alpha _{n,C}$ factors as a composition
\[ \mathscr {F}_{n}(C) \xrightarrow { \alpha '_{n,C} } \mathscr {F}'_{n}(C) \xrightarrow { \alpha ''_{n,C} } \mathscr {G}(C), \]
where $\alpha '_{n,C}$ is a monomorphism and $\alpha ''_{n,C}$ is a trivial Kan fibration (beware that $\mathscr {F}'_{n}(C)$ does not depend functorially on $C$). Form a pushout diagram
\[ \xymatrix@R =50pt@C=50pt{ \coprod _{C \in \operatorname{\mathcal{C}}} \underline{ \mathscr {F}_ n(C) } \times h^{C} \ar [r] \ar [d] & \coprod _{C \in \operatorname{\mathcal{C}}} \underline{ \mathscr {F}'_ n(C) } \times h^{C} \ar [d] \\ \mathscr {F}_{n} \ar [r] & \mathscr {F}_{n+1} } \]
in the category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})_{ / \mathscr {G } }$, where the upper horizontal map is the coproduct of the projective cofibrations described in Example 7.5.9.10. Using Remark 7.5.9.11, we see that each of the maps
\[ \mathscr {F}_{0} \rightarrow \mathscr {F}_{1} \rightarrow \mathscr {F}_{2} \rightarrow \mathscr {F}_{3} \rightarrow \cdots \]
is a projective cofibration. Setting $\mathscr {F} = \varinjlim _{n} \mathscr {F}_{n}$, we obtain a factorization of $\alpha _{0}$ as a composition $\mathscr {F}_{0} \xrightarrow { \beta } \mathscr {F} \xrightarrow { \alpha } \mathscr {G}$, where $\beta $ is a projective cofibration. We complete the proof by observing that for each object $C \in \operatorname{\mathcal{C}}$, the morphism $\alpha _{C}: \mathscr {F}(C) \rightarrow \mathscr {G}(C)$ is a trivial Kan fibration, since it can be written as a filtered colimit (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$) of the trivial cofibrations $\alpha ''_{n,C}: \mathscr {F}'_{n}(C) \rightarrow \mathscr {G}(C)$ (see Remark 1.5.5.3).
$\square$