  We develop a new perspective on principal bundles with connection as
morphisms from the tangent bundle of the underlying manifold to a classifying
dg-Lie groupoid. This groupoid can be identified with a lift of the inner
homomorphisms groupoid arising in \v{S}evera's differentiation procedure of Lie
quasi-groupoids. Our new perspective readily extends to principal groupoid
bundles, but requires an adjustment, an additional datum familiar from higher
gauge theory. The resulting adjusted connections naturally provide a global
formulation of the kinematical data of curved Yang-Mills-Higgs theories as
described by Kotov-Strobl (arXiv:1510.07654) and Fischer (arXiv:2104.02175).
