Consider three exact sequences of modules over a commutative ring
$0\to A\to M\to N\to B\to 0$,
$0\to E\to M\to K\to F\to 0$,
$0\to C\to N\to K\to D\to 0$,
where the maps $M\to N$ and $N\to K$ and $M\to K$ commute. I believe that diagram chasing gives that there is an exact sequence
$0\to A\to E\to C\to B\to F\to D\to 0$
that commutes with the maps in the other sequences. Is it possible to prove the exactness of this sequence from the standard lemmas in homological algebra?
Also, this appears to be a fairly simple result in that it states that for any commuting triangle, there is an exact sequence involving the kernels and cokernels of the three maps. Does this result have a name or is there a reference for this result?