A groupoid is a category in which every arrow has an inverse. The ESN (Ehresmann-Schein-Nambooripad) theorem states that the category of inverse semigroups is isomorphic to the category of inductive groupoids, which are groupoids with additional order-theoretic structure. Inductive groupoids are special types of ordered groupoids; the latter shares most of the properties of inductive groupoids but do not correspond to any type of semigroup. Despite this, many of the main facts about inverse semigroups carry over to ordered groupoids, and ordered groupoids have been shown to be an important tool in the study of inverse semigroups. Since then, it has been found that a particular type of partial algebra we call D-inverse constellations are equivalent to ordered groupoids, but have an arguably simpler, purely algebraic definition. This thesis will explore and illustrate the value of working with D-inverse constellations rather than ordered groupoids, presenting an alternative formulation and proof of the ESN Theorem using D-inverse constellations.