Lemma 6.45.
Let \(\phi^*\colon S \to T\) be a morphism of logoi and let \(U_\bullet\) be a hypercover in \(S\). Then \(\phi^*(U_\bullet)\) is a hypercover in \(T\).
Proof
Since \(\phi^*\) is left exact and the matching object \((\cosk_{n-1} U_\bullet)_n\) is a finite limit (Remark 6.43), we have \(\phi^*((\cosk_{n-1} U_\bullet)_n) \simeq (\cosk_{n-1} \phi^*(U_\bullet))_n\). Since \(\phi^*\) preserves effective epimorphisms, the result follows.