  The aim of this paper is to provide a definition of groupoid and cogroupoid
internal to a category which makes use of only one object and morphisms, in
contrast with the two object approach commonly found in the literature. We will
give some examples and we will establish a relation with group objects (and
Hopf algebras). The definitions presented here were designed to simplify some
constructions related to internalizations, Lie groupoids and Hopf algebroids.
