Full-faithfulness criterion for presheaf geometric embeddings

The geometric morphism induced by $T:\mathcal{C}\to \mathcal{D}$ is a geometric embedding exactly when $T$ is full and faithful. Its right adjoint has $(f_{\ast }H)(d)=\mathrm{Nat}(\mathcal{D}(T-,d),H)$. The counit at $c$ is precomposition with $\mathcal{C}(-,c)\to \mathcal{D}(T-,Tc)$. It is invertible for all $H$ precisely when that comparison presheaf is an isomorphism, by the Yoneda lemma, which is precisely full faithfulness of $T$.