APS homomorphism

This article gives a basic definition in the following area: APS theory
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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:

$f_{m+n}(\Phi _{m,n}(g,h))=\Psi _{m,n}(f_{m}(g),f_{n}(h))$