generalized element

Thinking of x∈_AB as a generalized element, for any f:B→C, we may write f(x) for the composite f∘x. In this notation the first domain–codomain axiom reads as follows: for any x∈_AB and f:B→C there is a well-defined f(x)∈_AC; that is, at each stage A, f takes A-elements of B to A-elements of C.