Proof.
Let \(\mathcal V_{Q/^{\mathcal L}}\) denote the full subcategory of \(\mathcal V_{Q/}\) on the morphisms \(Q\rightarrow X\) which are in \(\mathcal L\). The factorization system induces a right adjoint of the inclusion \(\mathcal V_{Q/^{\mathcal L}} \hookrightarrow \mathcal V_{Q/}\) which sends \(Q\rightarrow X\) to its factorization \(Q\rightarrow X|_Q\). The statement then follows immediately from adjunction. ◻