Categorical resolutions

It was observed in [Recent Progress in General Topology, North-Holland, 1992, pp. 637–757] that the bonding maps used in inverse limit systems, such as Mardešićʹs construction [Proc. Amer. Math. Soc. 11 (1996) 929–937] of the counterexample to the conjecture of the Hahn–Mazurkiewicz Theorem in non-metric spaces, can be replaced by the more flexible and intuitive resolutions. The construction of the counterexample used the idea of a pull-back. We show that some categorical resolutions imply a pull-back and other map notions under certain conditions.