The context is two statements from this paper:
A quasi-local ring $(R, M \cap R)$ contained in a complete local ring $(T,M)$ is Noetherian and has completion $T$ provided the map $R \to T/M^2$ is surjective and finitely generated ideals of $R$ are closed in the topology induced from $T$.
This fact is stated without explanation, and I couldn't track it down in Matsumura or the other references. It appears to be used in the final line of the main theorem (Theorem 8), which simply concludes
Finally, $A \to T/M^2$ onto implies $\widehat{A} \to T$ is surjective.
(Here, $A$ is playing the same role as $R$ above).
I haven't been able to figure out why this is true: in particular, why does the map $R \to T/M^2$ give us information about the $\widehat R \to \widehat T$? But my understanding of completions is fairly shallow, so it's possible this is a simple enough fact that the author felt it was self-evident.
I tracked down a reference in a later paper by the author, Completions of Local Rings with an Isolated Singularity.
Proposition 1 is exactly the statement above, and is presented with proof this time.