This is the part (c) in Exercise 5.2.4 of Luis Ribes' and Pavel Zalesskii's book Profinite Groups.
Let $\Lambda$ be a profinite ring, $(X, *)$ be a pointed profinite space and $(Y_i,*)$ be a family indexed by $i \in I$ of pointed closed subsets of $(X,*)$. Set $Y = \bigcap_{i \in I} Y_i$. Prove that, after identifying the free profinite $\Lambda$-modules below with their respective images in $[[\Lambda(X,*)]]$, we have the identity: $$[[\Lambda(Y,*)]] = \bigcap_{i \in I} [[\Lambda(Y_i,*)]]\,.$$
I've managed to prove this result assuming that the family $Y_i$ is filtered from below, i.e., for any pair $Y_i$ and $Y_k$ there exists a $Y_j$ satisfyng $Y_j \subseteq Y_i\cap Y_k$. This condition implies that the intersection commutes with continuous functions by a compactness argument. Is this exercise still true without the filtration hypothesis? I fail to see how I may go around proving this, but I haven't been able to provide a counter-example either.
EDIT My proof for the filtered case is also wrong. So far, all I know is that the free profinite $\Lambda$-module functor is exact in the category of pointed topological spaces.