Let $\mathcal{C}$ be a cofibrantly generated simplicial model category, say with its set of cofibrations $\beta$-saturated and generated by a set of $\beta$-small morphisms $\mathcal{M}_0$, and $I$ a small category.
For $i\in I$ we have adjunctions $F_i: \mathcal{C} \to \mathcal{C}^I: G_i$ by \begin{align*} G_iX = X(i); && \& && F_iD(j) = \bigsqcup_{I(i,j)} D \end{align*} for $X \in \mathcal{C}^I, D \in \mathcal{C}, j\in I$, with obvious action on morphisms.
One can show:
- If $A \to B \in \mathcal{M}_0$, then $F_iA \to F_iB$ is a pointwise cofibration in $\mathcal{C}^I$, i.e. all $G_jF_iA \to G_jF_iB$ are cofibrations in $\mathcal{C}$
- $\mathcal{M} := \{ \text{pointwise cofibrations of } \mathcal{C}^I \}$ is $\beta$-saturated
Hence it follows that the $\beta$-saturation of $\{F_iA \to F_iB \mid i\in I, A \to B \in \mathcal{M}_0\}$ is contained in $\mathcal{M}$. My question is, do we have equality and why? Goerss & Jardine claim we do ( Exmpl. II.6.9), but I don't see why.