[002D] Remark 2.3·d (Two ways to generalize pullbacks).

Hypocartesian [002A] and cartesian [0001] morphisms can be thought of as two distinct ways to generalize the concept of a pullback, depending on what one considers the essential properties of pullbacks. Hypocartesian morphisms more directly generalize the “little picture” universal property of pullbacks as limiting cones, whereas cartesian morphisms generalize the “big picture” dynamics of the pullback pasting lemma. As we have seen in Lemma 2.3·b [002C] these two notions coincide in any cartesian fibration; the instance of this result for the canonical self-indexing (Construction 2.1·a [001X]) verifies that pullbacks can be equivalently presented in terms of cartesian morphisms, as we have pointed out in Exercise 2.1·b [001Y].