I'm reading Sophie Morel's lecture notes on homological algebra.The Proposition V.2.1.6,p156 asserts：
Let $G:\mathcal{C}\to\mathcal{D}$ be a full and essentially surjective functor，$W$ be a set of morphisms of $\mathcal{C}$. Supposed that there exists a set of morphisms $W_{1}$ of $\mathcal{D}$ such that $W=\left \{s\in Mor\left(\mathcal{C}\right)\mid G\left(s\right)\in W_{1}\right \}$. Let $Q_{1}: \mathcal{D}\to \mathcal{D}\left[W_{1}^{-1}\right]$ be a localization of $\mathcal{D}$ by $W_{1}$, then $Q_{1}\circ G:\mathcal{C}\to \mathcal{D}\left[W_{1}^{-1}\right]$ is a localization of $\mathcal{C}$ by $W$.
The lecture notes has shown two properties of the three in the definition of localization of categories. Namely, $Q_{1}\circ G$ sends morphisms in $W$ to isomorphisms in $\mathcal{D}\left[W_{1}^{-1}\right]$; and, let$\mathcal{C}^{\prime}$be another category and $F_{1},F_{2}:\mathcal{D}\left[W_{1}^{-1}\right]\to \mathcal{C}^{\prime}$be functors, the natural map $Hom_{Func\left(\mathcal{D}\left[W_{1}^{-1}\right],\mathcal{C}^{\prime}\right)}\left(F_{1},F_{2}\right)\simeq Hom_{Func\left(\mathcal{C},\mathcal{C}^{\prime}\right)}\left(F_{1}\circ Q_{1}\circ G,F_{2}\circ Q_{1}\circ G\right)$ is bijective. But the notes doesn't show that $Q_{1}\circ G:\mathcal{C}\to \mathcal{D}\left[W_{1}^{-1}\right]$ satisfies the universal property of localization of categories.
My idea is, Let $Q:\mathcal{C}\to \mathcal{C}\left[W^{-1}\right]$ be a localization of $\mathcal{C}$ by $W$. Then we have a funtor $F_{W}:\mathcal{C}\left[W^{-1}\right]\to\mathcal{D}\left[W_{1}^{-1}\right]$，and a natural isomorphism $F_{W}\circ Q\simeq Q_{1}\circ G$, Then it suffices to show that $F_{W}$ is an equivalence of categories. But I don't know how to show $F_{W}$ is an equivalence of categories.
I feel that $W=\left \{s\in Mor\left(\mathcal{C}\right)\mid G\left(s\right)\in W_{1}\right \}$ is an important and nontrivial assumption, but I don't know how to use it.
Thanks for help!