Representably admissible sheaf maps

A map \(f\colon Y\to X\) in \(\widetilde C\) is representably admissible if, for every map \(h_U\to X\) from a representable sheaf, the pullback \(Y\times_Xh_U\) is representable and the induced morphism to \(h_U\) comes from a morphism in \(C^{\ad}\). We write \(\widetilde C^{\ad}_0\) for the resulting admissibility structure.