Corollary 2.49.
Let \(\Sigma\) be a local class in \(T\). Then there exists a classifying object for \(\Sigma\) if and only if for every object \(X \in T\) the full subanima of \((T_{/X})^{\simeq}\) spanned by the morphisms in \(\Sigma\) is small.
Proof
If a classifying object exists, then for every object \(X \in T\) the subanima of \((T_{/X})^{\simeq}\) spanned by the morphisms in \(\Sigma\) is equivalent to the small anima \(\Hom_T(X,X_{\Sigma})\), hence is itself small.Conversely, if for every object \(X \in T\) the subanima of \((T_{/X})^{\simeq}\) spanned by the morphisms in \(\Sigma\) is small, then these subanimae determine a functor \(T\catop \to \An\). By Proposition 2.47, this functor preserves limits. The adjoint functor theorem then implies it is represented by some object \(X_{\Sigma}\) of \(T\).