= 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 $(f_*H)(d)=\operatorname{Nat}(\mathcal D(T-,d),H)$. 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 $T$.
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