  In this article functorial Feynman rules are introduced as large
generalizations of physicists Feynman rules, in the sense that they can be
applied to arbitrary classes of hypergraphs, possibly endowed with any kind of
structure on their vertices and hyperedges. We show that the reconstruction
conjecture for classes of (possibly structured) hypergraphs admit a
sheaf-theoretic characterization, allowing us to consider analogous
conjectures. We propose an axiomatization for the notion of superposition
principle and prove that the functorial Feynman rules work as a bridge between
reconstruction conjectures and superposition principles, meaning that a
conjecture for a class of hypergraphs is satisfied only if each functorial
Feynman rule defined on it induces a superposition principle. Applications in
perturbative euclidean quantum field theory and graph theory are given.
