There is a fairly careful discussion of this in my book "Foundations of Global Non=Linear Analysis", Benjamin & Co. 1968
(Added later:) It occurred to me that this old book of mine is probably not easy to come by, so I have made a photo-copy of it available here:
http://vmm.math.uci.edu/FGNLA.pdf
I should add that there is also a quite detailed discussion of this topic in Chapters IX and X of my "Seminar on the Atiyah-Singer Index Theorem", Princeton Univ. Press, 1965. Unfortunately, I think that is still under copyright, so I cannot make it available, but on the other hand it should be readily available in almost any math library.