Let $K/F$ be Galois. I am trying to understand the construction of the Galois group for $K/F$ using inverse limits. To do this, we consider those fields $L$ such that $F\subset L\subset K$ and $L/F$ is finite and Galois. For intermediary fields $L$ we get a poset by inclusion (and, in fact, a directed set). Whenever $L_1\subset L_2$ we have the restriction map Gal$(L_2/F)\rightarrow$ Gal$(L_1/d)$ and so it makes sense to talk about the inverse limit.
Now, Gal$(K/F)$ is defined to be $\lim_L$Gal(L/F)$ where the inverse limit is taken over the directed set described above. However, since there may be multiple chains in this poset, does is matter which one we choose in the inverse limit? There may be no particular inclusion relation between two particular subfields, so I'm not sure what it would mean to take the inverse limit over all these fields.