Orbit category of a fusion system

Definition
Suppose $$P$$ is a group of prime power order and $$\mathcal{F}$$ is a defining ingredient::fusion system on $$P$$. The orbit category of $$\mathcal{F}$$ is defined as the following category:


 * Its objects are the subgroups of $$P$$.
 * The morphisms between two objects $$Q$$ and $$R$$ are the orbits in the set $$\operatorname{Hom}(Q,R)$$ under the action of the group of inner automorphisms of $$R$$ (denoted $$\operatorname{Aut}_R(R)$$ by post-composition. In other words, two elements $$\alpha,\beta: Q \to R$$ are in the same equivalence class iff there exists $$g \in R$$ such that for all $$x \in Q$$, $$\beta(x) = g\alpha(x)g^{-1}$$.

Note that the orbit category is not a concrete category because its morphisms are equivalence classes of maps rather than actual maps.