Definition 2.48.
Let \(\Sigma\) be a local class in \(T\). We say that a morphism \(f_{\Sigma}\colon Y_{\Sigma} \to X_{\Sigma}\) classifies \(\Sigma\) if for every object \(X \in T\) the map
\[\Hom_T(X,X_{\Sigma}) \to (T_{/X})^{\simeq}, \qquad g \mapsto g^*(f_{\Sigma})\]
is a monomorphism of animae whose image is given by the full subanima spanned by the morphisms in \(\Sigma\). In this situation, we also say that the object \(X_{\Sigma}\) classifies \(\Sigma\).