On the normal functor in the category of smooth vector bundles

This article is dedicated to the study of the normal functor in the category of smooth real vector bundles. Particularly, we focus on a symmetry phenomena which occurs after iterating two times the normal functor on a commutative square of smooth immersions. To do so, a theory of pullback and quotient is developed for double vector bundles but also for some classes of morphisms. These two operations turn out to be the key ingredients in order to study the naturality of the normal functor. The expected symmetry is then obtained thanks to the universal behavior and the mutual compatibility of these operations.