In that paper https://web.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf Hopkins-Kuhn-Ravenel introduced the idea of generalized character corresponding to a complex-oriented cohomology theory $E^*$. They construct an $E^*$-algebra $L(E^*)$, which is defined to be the colimit: ${\rm colim} E^*(B\mathbf{Z}^n_p)$ (More details regarding the whole construction can be found in the link.)

Later on, they use this construction to define the invariant ring $L(E^*)^{{\rm Aut}(\mathbf{Z}^n_p)}$, where ${\rm Aut}(\mathbf{Z}^n_p)$ acts as $E^*$-algebra homomorphisms ($\mathbf{Z}_p$ denotes the additive group of $p$-adic integers). To prove this they define $L_r(E^*)=E^*(B\mathbf{Z}_{p^r})$, and the natural ${\rm Aut}(\mathbf{Z}_{p^r})$ action on gives $L_r(E^*)^{{\rm Aut}(\mathbf{Z}_{p^r})}=p^{-1}E^*$.

My question has to do with the proof of the above: What I understand is that the invariant rings $L_r(E^*)^{{\rm Aut}(\mathbf{Z}_{p^r})}=p^{-1}E^*$, induce a direct system of $E^*$-algebras and the colimit must be $L(E^*)^{{\rm Aut}(\mathbf{Z}^n_p)}$. However, they don't give any proof hence should be somehow straightforward why the colimit is the invariant ring $L(E^*)^{{\rm Aut}(\mathbf{Z}^n_p)}$ (not at all to me). Can you explain me please if my understanding makes sense? If yes, probably an explanation why the above colimit converges on the invariant ring $L(E^*)^{{\rm Aut}(\mathbf{Z}^n_p)}$ would be really helpful. If not, a sort of insight would be very appreciable!