Quasi Left Factorization Structures as Presheaves

In this article the notion of quasi left factorization structure in a category ${\mathcal{X}}$ is given. It is proved that quasi left factorization structures correspond to subobjects of a predefined object in the category of laxed preordered valued presheaves, $\boldsymbol{L}ax({PrOrd}^{{\boldsymbol{\mathcal{X}}}^{op}})$ . The main result proves this correspondence is one to one when quasi left factorization structures are restricted to the so called QLF-codomains.