 The idea of using an initial property is to set up the problem in terms of some auxiliary category "E", so that the problem at hand corresponds to finding an initial object of "E".This has an advantage that the "optimization"—the sense that the process finds the "most efficient" solution—means something rigorous and is recognisable, rather like the attainment of a supremum.The category "E" is also formulaic in this construction, since it is always the category of elements of the functor to which one is constructing an adjoint.