Corollary 6.26.
Let \(v\colon (C,\tau) \to (D,\tau')\) be a functor between sites and assume that \(v\) admits a right adjoint \(u\colon D \to C\). Then \(v\) is a cocontinuous morphism of sites if and only if \(u\) is a continuous morphism of sites. In this case, we have \(v_* \simeq u^*\).
Proof
The adjunction \(v \dashv u\) induces an adjunction \(v^* \dashv u^*\) on presheaf categories, showing that \(v_* \simeq u^*\). By definition, \(v\) is a cocontinuous morphism of sites if and only if \(v_*\) preserves sheaves, while it follows from Remark 6.19 that \(u\) is a continuous morphism of sites if and only if \(u^*\) preserves sheaves, showing the claim.