APS homomorphism

Definition
Let $$(G,\Phi)$$ and $$(H,\Psi)$$ be APSes of groups. A homomorphism $$f:G \to H$$ is the following data: for every $$n$$, a homomorphism of groups $$f_n:G_n \to H_n$$, such that:

$$f_{m+n}(\Phi_{m,n}(g,h)) = \Psi_{m,n}(f_m(g),f_n(h))$$