Fusion system induced by a finite group on its p-Sylow subgroup is functorial

Statement
Suppose $$p$$ is a prime number. Consider the map from the category of finite groups to the category of fusion systems that sends a finite group to the fusion system induced on its p-Sylow subgroup. This map is functorial. In particular:


 * A homomorphism of groups induces a morphism of fusion systems.
 * The morphism of fusion systems induced by the identity morphism is the identity morphism.
 * The morphism of fusion systems induced by a composite of two homomorphisms is the composite of the morphisms induced by each of them.