Categorically, what's a limit is an object of ordered pairs (assuming you define them negatively, by their projections), not individual ordered pairs, right? I

Categorically, what's a limit is an object of ordered pairs (assuming you define them negatively, by their projections), not individual ordered pairs, right? If I recall correctly, categorical thinking doesn't emphasize elements of objects much, but in certain categories (perhaps pretoposes? I forget), morphisms from the terminal object to X can be considered the “global elements” of X.