  Motivated by the definition of the smooth manifold structure on a suitable
mapping space, we consider the general problem of how to transfer local
properties from a smooth space to an associated mapping space. This leads to
the notion of smoothly local properties.
  In realising the definition of a local property at a particular point it may
be that there are choices that need to be made. To say that the local property
is smoothly local is to say that those choices can be made smoothly dependent
on the point. In particular, that a manifold has charts is a local property. A
local addition is the structure needed to say that there is a way to choose a
chart about each point so that it varies smoothly with that point.
  To be able to extend these ideas beyond that of the local additivity of
manifolds we work in a category of generalised smooth spaces. We are thus are
able to consider more general mapping spaces than just those arising from the
maps from one smooth manifold to another and thus able to generalise the
standard result on when this space of maps is again a smooth manifold.
  As applications of this generalisation we show that the mapping spaces
involving the various figure 8s from String Topology are manifolds, and that
they embed as submanifolds with tubular neighbourhoods in the corresponding
loop spaces. We also show that applying the mapping space functor to a regular
map of manifolds produces a regular map on the mapping spaces.
