APS-on-APS action

Definition
Let $$(G,\Phi)$$ be an APS of groups and $$(T,\Psi)$$ an APS of sets. An action of $$G$$ on $$T$$ is defined as the data of an action of $$G_n$$ on $$T_n$$ for every $$n$$, such that for any natural numbers $$m,n$$ and any elements $$g \in G_m, h \in G_n, s \in T_m, t \in T_n$$, we have:

$$\Phi_{m,n}(g,h) \cdot \Psi_{m,n}(s,t) = \Psi_{m,n}(g \cdot s, h \cdot t)$$