Remark 4.4.
By the adjoint functor theorem, every morphism of logoi \(\phi^*\colon S \to T\) admits a right adjoint \(\phi_*\colon T \to S\), which is then a morphism of topoi. This assignment can be made functorial: there is an equivalence of categories
\[\Topos \quad \simeq \quad \Logos\catop, \qquad T \mapsto T, \qquad \phi_* \mapsto \phi^*.\]
More precisely, this equivalence arises as a restriction of the equivalence \(\PrR \simeq (\PrL)\catop\) from [Lurie 2009, Corollary 5.5.3.4].
References
- Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.