definition. geometric morphism, Topoi [kostecki2011introduction, 7.2] [tt-004Q]

✍️source

If \({\cal E}_1\) and \({\cal E}_2\) are toposes, then a geometric morphism \(\mathscr {G}: {\cal E}_1 \to {\cal E}_2\) is defined as a pair of adjoint functors \(\mathscr {G}^* \dashv \mathscr {G}_*\) between \({\cal E}_1\) and \({\cal E}_2\), such that \(\mathscr {G}^*\) preserves finite limits (i.e. is left exact), which implies \(\mathscr {G}_*\) preserves colimits (i.e. is right exact).

The category of toposes and their geometric morphisms is denoted \(\mathbf {Topoi}\).