We provide proofs of some key propositions that were used in previous work by Dossa and Tadmon dealing with the characteristic initial value problem for the Einstein-Yang-Mills-Higgs system. The aforesaid proofs were missing, making the considered work difficult to understand. The work is presented with a view to have an almost self-contained paper. With this respect we completely recall the process of constructing initial data for the Einstein-Yang-Mills-Higgs system on two intersecting smooth null hypersurfaces as done in the work of Dossa and Tadmon mentioned above. This is achieved by successfully adapting the hierarchical method set up by Rendall to solve the same problem for the Einstein equations in vacuum and with perfect fluid source. Unlike the work of Rendall, many delicate calculations and expressions are given in details so as to address, in forthcoming work, the issue of global resolution of the characteristic initial value problem for the Einstein-Yang-Mills-Higgs system. The method obviously applies to the Einstein-Maxwell and Einstein-scalar field models as well.