 Geometric morphism

A geometric morphism $f:\mathcal{E}\to \mathcal{F}$ is an adjunction $f^{\ast }⊣f_{\ast }$ with finite-limit-preserving inverse image $f^{\ast }:\mathcal{F}\to \mathcal{E}$. Inverse image also preserves colimits as a left adjoint. An extra left adjoint $f_{!}⊣f^{\ast }$ makes it essential.