  We study the homotopy right Kan extension of homotopy sheaves on a category
to its free cocompletion, i.e. to its category of presheaves. Any pretopology
on the original category induces a canonical pretopology of generalised
coverings on the free cocompletion. We show that with respect to these
pretopologies the homotopy right Kan extension along the Yoneda embedding
preserves homotopy sheaves valued in (sufficiently nice) simplicial model
categories. Moreover, we show that this induces an equivalence between sheaves
of spaces on the original category and colimit-preserving sheaves of spaces on
its free cocompletion. We present three applications in geometry and topology:
first, we prove that diffeological vector bundles descend along subductions of
diffeological spaces. Second, we deduce that various flavours of bundle gerbes
with connection satisfy $(\infty,2)$-categorical descent. Finally, we
investigate smooth diffeomorphism actions in smooth bordism-type field theories
on a manifold. We show how these smooth actions allow us to extract the values
of a field theory on any object coherently from its values on generating
objects of the bordism category.
