Corollary 11.6.0.8. See Proposition 6.3.2.4.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Apply Proposition 6.3.4.2 to the cubical diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [dr] \ar [rr] \ar [dd] & & \operatorname{\mathcal{C}}' \ar [dd] \ar [dr] & \\ & \operatorname{\mathcal{C}}\ar [rr] \ar [dd] & & \operatorname{\mathcal{C}}' \ar [dd] \\ \operatorname{\mathcal{C}}\ar [rr] \ar [dr] & & \operatorname{\mathcal{C}}' \ar [dr] & \\ & \operatorname{\mathcal{D}}\ar [rr] & & \operatorname{\mathcal{D}}'. } \]
$\square$