Saturated fusion system

Definition
Let $$P$$ be a group of prime power order, say a finite $$p$$-group, for a prime $$p$$. A saturated fusion system $$\mathcal{F}$$ on $$P$$ is a category on $$P$$ (in the sense of a category on a finite p-group) with the following properties:

Terminological note
Some people use the term fusion system for what is defined here as a saturated fusion system. Others use fusion system to mean a category on a finite p-group that contains the inner fusion system. People in both terminological camps are usually interested only in saturated fusion systems when they talk of fusion systems.