Definition 6.85. (Locally of constant shape)
Let \(\phi_*\colon T \to S\) be a morphism of topoi. Given an object \(U \in T\), we say that \(U\) has constant shape relative to \(\phi_*\) if the composite morphism
\[T_{/U} \xrightarrow{j_*} T \xrightarrow{\phi_*} S\]
is of constant shape.
We say that \(\phi_*\) is locally of constant shape if there exists a collection of objects \(\{U_i\}_{i \in I}\) which generates \(T\) under colimits such that each \(U_i\) has constant shape relative to \(\phi_*\).
A topos \(T\) is locally of constant shape if the terminal morphism \(\Gamma_*\colon T \to \An\) is locally of constant shape.