Geometric morphism
= Geometric morphism
{title2=$f^*\dashv f_*$}
A geometric morphism $f:\mathcal E\to\mathcal F$ is an <adjunction> $f^*\dashv f_*$ with finite-limit-preserving inverse image $f^*:\mathcal F\to\mathcal E$. Inverse image also preserves colimits as a left adjoint. An extra left adjoint $f_!\dashv f^*$ makes it essential.