[Theorem 8.1, Ma]
Lemma 85.4.5. Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$-module. Let $N \subset M$ be a submodule. Assume $M$ has a countable fundamental system of neighbourhoods of $0$. Then
$0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge \to 0$ is exact,
$N^\wedge $ is the closure of the image of $N \to M^\wedge $,
$M^\wedge \to (M/N)^\wedge $ is open.
We have $0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge $ is exact and statement (2) by Lemma 85.4.3. This produces a canonical map $c : M^\wedge /N^\wedge \to (M/N)^\wedge $. The module $M^\wedge /N^\wedge $ is complete and $M^\wedge \to M^\wedge /N^\wedge $ is open by Lemma 85.4.4. By the universal property of completion we obtain a canonical map $b : (M/N)^\wedge \to M^\wedge /N^\wedge $. Then $b$ and $c$ are mutually inverse as they are on a dense subset.