 The monics of a subobject will in general have many domains, all of which however will be in bijection with each other.To summarize, this first-order notion of subobject classifier implicitly defines for a topos the same equivalence relation on monics to "X" as had previously been defined explicitly by the second-order notion of subobject for any category.The notion of equivalence relation on a class of morphisms is itself intrinsically second-order, which the definition of topos neatly sidesteps by explicitly defining only the notion of subobject "classifier" Ω, leaving the notion of subobject of "X" as an implicit consequence characterized (and hence namable) by its associated morphism "f"