Logoi and morphisms of logoi

A category \(T\) is called a logos if it is a topos, i.e. if it is presentable and satisfies descent for all colimits. Given two logoi \(S\) and \(T\), a morphism of logoi (sometimes called algebraic morphism\footnote{We warn the reader that this terminology clashes with the notion of `algebraic morphism' used by Lurie (2009, Definition 6.3.6.1).}) from \(S\) to \(T\) is a left exact colimit-preserving functor \(\phi^*\colon S \longrightarrow T\). We denote by

the subcategory spanned by the logoi and the morphisms of logoi.