The locally free relatively filtered diagram as an inductive completion of a system of choice (Q1353445)

A functor \(U:{\mathcal A}\to{\mathcal B}\) admits locally free diagrams (L.F.Ds) if for any \(B\in{\mathcal B}\), there exists a category \({\mathcal L}FD_U(B)\), a functor \(D^B_U:{\mathcal L}FD_U(B)\to {\mathcal A}\) and a projective cone \(d^B_U= (B\to UD^B_U(C))_{C\in{\mathcal L}FD_U(B)}\) such that for any \(A\in{\mathcal A}\), we have \[ \text{Hom}_{\mathcal B}(B,U(A))\simeq\varinjlim_{C\in{\mathcal L}FD_U(B)}\text{Hom}_{\mathcal A}(D^B_U(C),A). \] It admits locally free relatively filtered diagrams (L.F.R.F.Ds) if moreover for any morphism \(h:B\to U(A)\) in \(\mathcal B\), the comma category \({\mathcal H}\downarrow h\) is filtered, where \(H:{\mathcal L}FD_U(B)\to (B\downarrow U)\) denotes the functor assigning \(d^B_U(C): B\to UD^B_U(C)\) to \(C\). The paper provides a new constructive proof for the existence of L.F.Ds in which the effective part and the non effective part are well separated, for some sketchable functor \(U\). Moreover, it shows that they are L.F.R.F.Ds.