Proof.
Assume that \(\mathcal C_{X_1} \times \cdots \mathcal C_{X_n} \rightarrow\mathcal C_{X}\) carries morphisms in \(S_{X_1} \cdots \times \cdots S_{X_n}\) to morphisms in \(\mathcal L_X\). By assumption, the functor \(\mathcal C_{X_1} \times \cdots \times \mathcal C_{X_n} \rightarrow\mathcal C_{X}\) preserves small colimits separately in all variables. Since for every \(Y \in \underline{\mathcal O}\), the class of morphisms \(\mathcal L_Y\) is by proposition B.1.14 the smallest saturated class of morphisms in \(\mathcal C_Y\) that contains \(S_Y\), the functor \(\mathcal C_{X_1} \times \cdots \times \mathcal C_{X_n} \rightarrow\mathcal C_{X}\) therefore also carries morphisms in \(\mathcal L_{X_1} \times \cdots \times \mathcal L_{X_n}\) to morphisms in \(\mathcal L_X\). ◻