Given a directed set $A$, a subset $B \subset A$ is said to be coinitial if for every $a \in A$ there is some $b \in B$ such that $b \leq a$.
Now consider an inverse system of rings $(S_i, f_{ij})$ indexed by $A$ and let $B \subset A$ be coinitial.
Is it true that $\varprojlim_{i \in A}S_i \cong \varprojlim_{i \in B} S_i$?
I am aware that the answer is true if $B$ is cofinal instead of coinitial. I as wondering if there is some sort of modification in the proof for cofinal sets that works as well for coinitial sets. Thank you in advance.