Forget the phase transfer for the moment.
1). Try to draw the mechanism of epoxidation of a double bond (alkene) by H2O2 in presence of an acid (acidic catalysis)
2). Now, replace the acidic proton by the tungsten Lewis acid and that’ all.
Getting back to phase transfer catalysis. The term “phase transfer” refers to the reaction conditions and not to the reaction catalysis. H2O2 and the tungsten phosphate catalyst are water soluble, while seed oil and seed oil rubber are not. So, the seed oil rubber is dissolved in and organic solvent (organic phase) and the rest in water (aqueous phase) and they are vigorously stirred together in the presence of a quaternary ammonium salt (C27H50N+, Cl-) that is a cationic surfactant and permits the interfacial contact of the two phases, where the reaction occurs. Phase transfer reactions are schematically represented by the attached reaction scheme.
The confusion starts by wrongly naming the cationic surfactant as a ”catalyst” for phase contact (= transfer, because reactants are transferred from the one phase to the other).