Right exactness of projective systems Suppose that we have a systems of exact sequences $(A_{n}\rightarrow B_{n}\rightarrow C_{n}\rightarrow 0)_{n\in \mathbb{N}}$ together with transitions maps $(A_{n+1}\rightarrow A_{n})_{n\in \mathbb{N}}, (B_{n+1}\rightarrow B_{n})_{n\in \mathbb{N}}, (C_{n+1}\rightarrow C_{n})_{n\in \mathbb{N}}$ making all the relevant diagrams commute. Suppose that the transition maps make $(A_{n})_{n\in \mathbb{N}},(B_{n})_{n\in \mathbb{N}},(C_{n})_{n\in \mathbb{N}}$ into systems that satisfy the Mittag-Leffler condition. Is it true that
$$\varprojlim_{n}A_{n}\rightarrow \varprojlim_{n}B_{n} \rightarrow \varprojlim_{n}C_{n}\rightarrow 0 \ \ (\ast)$$
is exact?
This is well-known to hold if we had short-exact sequences instead (and for that it would have been enough to only require that $(A_{n})_{n\in \mathbb{N}}$ satisfies the Mittag-Leffler condition), so I was wondering whether if we factor the above system into one of short exact sequences, then the factors will automatically satisfy the Mittag-Leffler condition too? Or is there some counterexample to $(\ast)$?
 A: Define an inverse system of short exact sequences \begin{align*} \textstyle S_{n} = \{0 \to K_{n} \to A_{n} \to B_{n} \to 0\} = \{0 \to \mathbb{Z} \to \mathbb{Z}[\frac{1}{p}] \to \mathbb{Z}[\frac{1}{p}]/\mathbb{Z} \to 0\} \end{align*} where all the vertical maps in the morphism $S_{n+1} \to S_{n}$ are multiplication-by-$p$. We have $\varprojlim_{n} K_{n} = 0$ since no nonzero integer is $p$-divisible, we have $\varprojlim_{n} A_{n} = \mathbb{Z}[\frac{1}{p}]$ since the transition maps $K_{n+1} \to K_{n}$ are isomorphisms, and $\varprojlim_{n} B_{n}$ contains $\mathbb{Z}_{p}$: take the limit of inclusions $\mathbb{Z}/p^{n}\mathbb{Z} \to \mathbb{Z}[\frac{1}{p}]/\mathbb{Z}$ sending $1 \mapsto \frac{1}{p^{n}}$. Since $\varprojlim_{n} A_{n}$ is countable and $\varprojlim_{n} B_{n}$ is uncountable, the morphism $\varprojlim_{n} A_{n} \to \varprojlim_{n} B_{n}$ is not surjective.
We define $C_{n} = 0$ for all $n$. The systems $\{A_{n}\},\{B_{n}\},\{C_{n}\}$ satisfy ML since all transition maps are surjective.
