generalized element

Noun

 * 1)  A morphism whose codomain is some specified object.
 * Thinking of $$x \in_A B$$ as a generalized element, for any $$f: B \rightarrow C$$, we may write $$f(x)$$ for the composite $$f \circ x$$. In this notation the first domain–codomain axiom reads as follows: for any $$x \in_A B$$ and $$f: B \rightarrow C$$ there is a well-defined $$f(x) \in_A C$$; that is, at each stage A, f takes A-elements of B to A-elements of C.
 * Thinking of $$x \in_A B$$ as a generalized element, for any $$f: B \rightarrow C$$, we may write $$f(x)$$ for the composite $$f \circ x$$. In this notation the first domain–codomain axiom reads as follows: for any $$x \in_A B$$ and $$f: B \rightarrow C$$ there is a well-defined $$f(x) \in_A C$$; that is, at each stage A, f takes A-elements of B to A-elements of C.