If anyone is interested in some more synthetic differential geometric goodness, the point of view of the book Natural Operations in Differential Geometry by Ivan Kolar, Jan Slovak and Peter W. Michor, while couched in a more traditional language, is quite close to that of synthetic differential geometry. In Natural Operations, the authors are trying to classify functors on the category of differentiable manifolds (this is what they call a natural operation). Synthetic differential geometry tries to define a larger category so that those functors become representable.