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 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 $\Phi(\varphi) \circ \alpha = \alpha \circ \varphi$ .