Geometric morphism ($f^{\ast }⊣f_{\ast }$)

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.

A topos over sets is local when its global sections functor is itself an inverse image functor. For an idempotent-complete small indexing category, its presheaf topos is local exactly when the indexing category has a terminal object: then global sections are evaluation at that object.

A functor $F:\mathcal{C}\to \mathcal{D}$ induces a geometric morphism from the presheaf topos on $\mathcal{C}$ to that on $\mathcal{D}$. Its inverse image is precomposition with $F^{\mathrm{op}}$, its direct image is Right Kan extension and its extra left adjoint is left Kan extension. The extra left adjoint sends $yC$ to $y(FC)$.