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Proposition 6.3.1.20 (Transitivity). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ and $F': \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}''$ be morphisms of simplicial sets. Let $W$ and $W'$ be collections of edges of $\operatorname{\mathcal{C}}$ satisfying the following conditions:

  • The morphism $F$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$.

  • The morphism $F'$ exhibits $\operatorname{\mathcal{C}}''$ as a localization of $\operatorname{\mathcal{C}}'$ with respect to $F(W')$.

Then the composite morphism $(F' \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}''$ exhibits $\operatorname{\mathcal{C}}''$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W \cup W'$.

Proof. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category; we wish to prove that precomposition with $F' \circ F$ induces an equivalence from $\operatorname{Fun}(\operatorname{\mathcal{C}}'', \operatorname{\mathcal{E}})$ to the full subcategory $\operatorname{Fun}( \operatorname{\mathcal{C}}[ (W \cup W')^{-1} ], \operatorname{\mathcal{E}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. We have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \operatorname{\mathcal{C}}'', \operatorname{\mathcal{E}}) \ar [r]^-{\circ F} & \operatorname{Fun}( \operatorname{\mathcal{C}}'[ F(W')^{-1} ], \operatorname{\mathcal{E}}) \ar [r]^-{\circ F'} \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}[ (W \cup W')^{-1} ], \operatorname{\mathcal{E}}) \\ & \operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{E}}) \ar [r]^-{\circ F'} & \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1} ], } \]

where the horizontal functors on the left and lower right are equivalences of $\infty $-categories. Since the square is a pullback and the vertical maps are isofibrations (Remark 6.3.1.8), it follows that the horizontal map on the upper right is also an equivalence of $\infty $-categories (Corollary 4.5.4.6). $\square$