Definition 2.41.
An object classifier is an object \(U \in \widehat{T} := \Fun^{\mathrm{lim}}(T\catop,\widehat{\An})\) such that \(\Hom(X,U) \simeq (T_{/X})^{\simeq}\). Note that \(T\) admits an object classifier if and only if the assignment \(X \mapsto (T_{/X})^{\simeq}\) preserves limits.