Morphism of fusion systems

Definition
Suppose $$P_1, P_2$$ are both finite $$p$$-groups (i.e., groups of prime power order for some prime number $$p$$). Suppose $$\mathcal{F}_1$$ is a defining ingredient::fusion system on $$P_1$$ and $$\mathcal{F}_2$$ is a fusion system on $$P_2$$. A morphism of fusion systems from $$\mathcal{F}_1$$ to $$\mathcal{F}_2$$ is a pair $$(\alpha,\Phi)$$ where:


 * $$\alpha$$ is a homomorphism of groups from $$P_1$$ to $$P_2$$.
 * $$\Phi$$ is a covariant functor from the category $$\mathcal{F}_1$$ to the category $$\mathcal{F}_2$$.
 * For every subgroup $$Q \le P$$, $$\alpha(Q) = \Phi(Q)$$. Here, $$\alpha(Q)$$ is the homomorphic image and $$\Phi(Q)$$ is the image of the object $$Q \in \mathcal{F}_1$$ under the covariant functor $$\Phi$$.
 * If $$\varphi:Q \to R$$ is a morphism in $$\mathcal{F}_1$$, then $$\Phi(\varphi) \circ \alpha = \alpha \circ \varphi$$.