  For a smooth (locally trivial) principal bundle in Ehresmann's sense, the
relation between the commuting vertical and horizontal actions of the
structural Lie group and the structural Lie groupoid (isomorphisms between
vertical fibers) is regarded as a special case of a symmetrical concept of
conjugation between "principal" Lie groupoid actions, allowing possibly
non-locally trivial bundles. A diagrammatic description of this concept via a
symmetric "butterfly diagram" allows its "internalization" in a wide class of
categories (used by "working mathematicians") whenever they are endowed with
two distinguished classes of monomorphisms and epimorphisms mimicking the
properties of embeddings and surjective submersions. As an application, a
general theorem of "universal activation" encompasses in a unified way such
various situations as Palais' theory of globalization for partial action laws,
the realization of non-abelian cocycles (including Haefliger cocycles for
foliations) or the description of the "homogeneous space" attached to an
embedding of Lie groups (still valid for Lie groupoids).
