I think exponentials and coexponentials are relevant here, since they are good at shuffling things back and forth between the sides of morphisms, which matters for limits and colimits as they are adjunctions (and a particularly nice kind of adjunction, at that).
I think exponentials and coexponentials are relevant here, since they are good at shuffling things back and forth between the sides of morphisms, which matters for limits and colimits as they are adjunctions (and a particularly nice kind of adjunction, at that).