I have a question regarding the proof of Proposition 2.19 of Factorization homology of topological manifolds by Ayala and Francis. In the final paragraph of the proof (more specifically, in the second sentence of the paragraph), they seem to make use of the following assertion:
($\ast$) Let $\mathcal{C}$ be an $\infty$-category and $\mathcal{W}\subset \mathcal{C}$ its subcategory. The localization map $\mathcal{C}\to\mathcal{C}[\mathcal{W}^{-1}]$ induces a weak homotopy equivalence $$\operatorname{Fun}(\Delta^1,\mathcal{C})\times_{\mathcal{C}\times \mathcal{C}}(\mathcal{W}\times \mathcal{W})\to\operatorname{Fun}(\Delta^1,\mathcal{C}[\mathcal{W}]^{-1})\times _{\mathcal{C}[\mathcal{W}^{-1}]\times \mathcal{C}[\mathcal{W}^{-1}]} (\mathcal{C}[\mathcal{W}^{-1}]^{\simeq}\times \mathcal{C}[\mathcal{W}^{-1}]^{\simeq}).$$
(If it's releveant at all, I should mention that they seem to be using assertion ($\ast$) only when $\mathcal{C}$ is the nerve of an ordinary category.) Here by localization, we mean that for each $\infty$-category $\mathcal{D}$, the functor $\operatorname{Fun}(\mathcal{C}[\mathcal{W}^{-1}],\mathcal{D})\to\operatorname{Fun}(\mathcal{C},\mathcal{D})$ is fully faithful and its essential image consists of the functors which maps every morphism in $\mathcal{W}$ to an equivalence of $\mathcal{D}$. Also, the superscript "$\simeq$" indicates maximal sub Kan complex.
Does anyone know how to prove ($\ast$)? Thanks in advance.
