Given a category C and a class of "weak equivalences" W in C, the morphisms in the localization Ho(C) of C at W generally consist of "zigzags" of maps in C and W of arbitrary length. In certain good cases, however, any morphism in Ho(C) is represented by a span of a leftward pointing weak equivalence followed by a rightward pointing map in C. It is known that these good cases yield a similar calculus of fractions in the context of infinity-categories and infinity-categorical localization. We aim to describe an alternative proof of this fact using a special kind of cocartesian fibration. If time permits, we describe how this approach gives a quick proof that the underlying infinity-category of a fibration category (resp. model of homotopy type theory with appropriate type constructors) has finite limits (resp. is locally cartesian closed).