Suppose $\require{AMScd}$ \begin{CD} @.\acute M @>f>> M @>g>>\check M @>>> O\\ @. @V \alpha V V\ @VV \beta V @VV \gamma V @. \\ O@>>> \acute N @>\phi>> N @>\omega>> \check N @. \end{CD} is commutative diagram from $R$-modules and $R$-homomorphisms, s.t. every row is exact sequence. How To Prove this row is exact sequence: $\require{AMScd}$ \begin{CD} Ker \alpha @>\check f>> Ker\beta @>>>Ker\gamma @>d>> \frac{\acute N}{Im \alpha} @>>> \frac{N}{Im \alpha} @>\check \omega >> \frac{\check N}{Im \alpha}\\ \end{CD} also : if $f$ is one-one function then $\check f$is one-one function,and if $\omega$ is onto function then $\check \omega$ is onto function.
it's my effort: we have: $Imf=Ker g=ker O$ ,$ImO=Ker \phi=Im\omega$ by (every rows is exact sequence)
$\phi \alpha=\beta f,,\omega \beta=\gamma g$ by (is commutative diagram) we must find 5 function s.t. $Im\phi_{i-1}=Ker\phi_i$ also we have "short five lemma" (perhaps)
The kernels go in a row above the diagram, and the cokernels go in the row below. The trickiest part of the exact sequence you need is in the middle. To construct the map $d$ one must watch this video:
https://www.youtube.com/watch?v=etbcKWEKnvg