  In this paper we prove an equivalence theorem originally observed by Robert
MacPherson. On one side of the equivalence is the category of cosheaves that
are constructible with respect to a locally cone-like stratification. Our
constructibility condition is new and only requires that certain inclusions of
open sets are sent to isomorphisms. On the other side of the equivalence is the
category of functors from the entrance path category, which has points for
objects and certain homotopy classes of paths for morphisms. When our
constructible cosheaves are valued in $\mathsf{Set}$ we prove an additional
equivalence with the category of stratified coverings.
