About Exact Sequences

Question

Suppose I have the following exact sequence:

$$\begin{matrix} A_0& \xrightarrow{\quad\quad} & B_0 \xrightarrow{\quad\alpha\quad} & C_0\\ \uparrow&&&\downarrow \\ C_1 & \xleftarrow{\quad\beta\quad}& B_1 \xleftarrow{\quad\quad}& A_1 \end{matrix}$$

Then I obtain two short exact sequence using $ \alpha$ and $ \beta$ ( Kernels and cokernels ) with $A_0$ and $A_1$ in the middle.

Any help as to how I get these sequences?

Thanks!

Answer 1

$$ 0\longrightarrow \operatorname{Coker} \beta\longrightarrow A_0\longrightarrow \operatorname{Ker\alpha}\longrightarrow0 $$ because, by the exactness of the original sequence, $\operatorname{Im}(A_0\to B_0)=\operatorname{Ker}\alpha$ and $\operatorname{Ker}(A_0\to B_0)=C_1/\operatorname{Im}\beta\equiv\operatorname{Coker}\beta$.

The second sequence is similar.