Let $C^n(G,A)$ be the set of continuous functions $G^n \rightarrow A$ with $G$ a profinite group and $A$ a discrete $G$-module (these are the functions that are locally constant). I want to prove that $C^n(G,A) = \varinjlim C^n(G/U,A^U)$, where $U$ runs through all open normal subgroups of $G$ and $A^U$ is the submodule fixed by $U$.
I think that the direct system you have to use is the following: let $U \subset V$ be 2 open normal subgroups of $G$. Then we have a canonical projection $p_{UV}:G/U \rightarrow G/V$, which is clearly continuous since both sides have the discrete topology. We also have a canonical inclusion $i_{UV}:A^V \rightarrow A^U$ which is again continuous because of the discrete topology on both sides. So we can make a function $\rho_{VU}:C^n(G/V,A^V) \rightarrow C^n(G/U,A^U)$, defined by $\rho_{VU}(\phi)=i_{UV}\phi p_{UV}$.
Now, I was trying to make an isomorphism from $C^n(G,A)$ to $\varinjlim C^n(G/U,A^U)$, but I don't see how to do this. I must admit, I'm not very good with limits in categories.
Any help would be appreciated.