Corollary 6.3.4.3 (03E3)

Corollary 6.3.4.3. Suppose we are given a categorical pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{F} \ar [r]^-{G} & \operatorname{\mathcal{C}}' \ar [d]^{F'} \\ \operatorname{\mathcal{D}}\ar [r] & \operatorname{\mathcal{D}}', } \]

where $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to some collection of edges $W$. Then $F'$ exhibits $\operatorname{\mathcal{D}}'$ as a localization of $\operatorname{\mathcal{C}}'$ with respect to $G(W)$.

Proof. Apply Proposition 6.3.4.2 to the cubical diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [dr] \ar [rr] \ar [dd] & & \operatorname{\mathcal{C}}' \ar [dd] \ar [dr] & \\ & \operatorname{\mathcal{C}}\ar [rr] \ar [dd] & & \operatorname{\mathcal{C}}' \ar [dd] \\ \operatorname{\mathcal{C}}\ar [rr] \ar [dr] & & \operatorname{\mathcal{C}}' \ar [dr] & \\ & \operatorname{\mathcal{D}}\ar [rr] & & \operatorname{\mathcal{D}}'. } \]

$\square$