  In this article we discuss local aspects of 2-functors defined on the path
2-groupoid of a smooth manifold; in particular, local trivializations and
descent data. This is a contribution to a project that provides an axiomatic
formulation of connections on (possibly non-abelian) gerbes in terms of
2-functors. The main result of this paper establishes the first part of this
formulation: we prove an equivalence between the globally defined 2-functors
and their locally defined descent data. The second part appears in a separate
publication; there we prove equivalences between descent data, on one side, and
various existing versions of gerbes with connection on the other side.
