Structure of finite dimensional linear bundles morphisms

Let $p_E:E\to M$ be a fibre bundle over the $m$-dimensional manifold $M$ whose typical fibre is the vector space ${\mathbb{R}}^e$ and let $p_F:F\to N$ be a fibre bundle over the $n$-dimensional manifold $N$ whose typical fibre is the vector space ${\mathbb{R}}^f$. We are interested in the structure of the set $\text{Morph}(E,F)$ of smooth linear bundle morphisms $\mathrm{\Phi }:E\to F$ above smooth maps $\phi :M\to N$.