Pullback-preserving APS

Definition
An APS of groups $$(G,\Phi)$$ is termed pullback-preserving if for any natural numbers $$m,n,p$$, the pullback of the pair of maps $$\Phi_{m+n,p}, \Phi_{m,n+p}$$ is the pair of maps $$\operatorname{id} \times \Phi_{n,p}, \Phi_{m,n} \times \operatorname{id}$$.