# Morphism of fusion systems

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