Warning 6.3.3.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. If $F$ and $G$ are localization functors, the composition $G \circ F$ need not be a localization functor. For example, let $\operatorname{\mathcal{C}}$ be the $1$-dimensional simplicial set corresponding to the directed graph depicted in the diagram
There is a functor $F: \operatorname{\mathcal{C}}\rightarrow \Delta ^2$ which carries $e$ and $e'$ to the edges $0 \rightarrow 1$ and $1 \rightarrow 2$, respectively. It is not difficult to show that $F$ exhibits $\Delta ^2$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $w$. However, the localization of $\Delta ^2$ with respect to its “long edge” $0 \rightarrow 2$ cannot be realized directly as a localization of $\operatorname{\mathcal{C}}$.