Let \(\mathcal M\) denote the category of smooth finite-dimensional manifolds and their smooth maps, and let \(\mathcal {FM}\) denote its fibred analogue. The main result of the paper is an equivalence between (1) product preserving bundle functors \(F : \mathcal {FM} \to \mathcal {FM}\), up to the natural equivalence of functors, and (2) triples \(\mu ,G,H\) where \(G,H\) are product preserving bundle functors \(\mathcal M \to \mathcal {FM}\) and \(\mu \) is a natural transformation between them, up to the equivalence of natural transformations. The paper ends with a corollary which characterizes vertical Weil bundle functors \(F : \mathcal {FM} \to \mathcal {FM}\).