Meaning of: any exact sequence'can be recovered by "composing" the short exact sequences...

Question

The following is taken from: $\textit{Abstract Algebra}$ by: P. A. Grillet

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$\textbf{Exercise:}$ Explain how any exact sequence $A\xrightarrow{\varphi}B\xrightarrow{\psi}C$ can be recovered by "composing" the short exact sequences $0\xrightarrow{}\text{Ker }\varphi \xrightarrow{}A\xrightarrow{}\text{Im }\varphi\xrightarrow{}0,$ $0\xrightarrow{}\text{Im }\varphi\xrightarrow{}B\xrightarrow{}\text{Im }\psi\xrightarrow{}C,$ and $0\xrightarrow{}\text{Im }\psi\xrightarrow{}C\xrightarrow{}C/\text{Im }\psi\xrightarrow{}C.$

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Can someone explain to me what Grillet meant by asking the reader to show that any exact sequence'can be recovered by "composing" the short exact sequences....'

Thank you in advance

Answer 1

Subject to correcting the error pointed out in my comments:

• You recover $A$, $B$, $C$ from the middle terms of the 1st, 2nd and 3rd short exact sequences respectively;
• You recover the homomorphism $A \xrightarrow{\varphi} B$ by composing the two homomorphisms $A \mapsto \text{Im} \, \phi \mapsto B$ from the 1st and 2nd short exact sequences respectively;
• And you recover the homomorphism $B \xrightarrow{\psi} C$ by composing the two homomorphisms $B \mapsto \text{Im} \, \psi \mapsto C$ from the 2nd and 3rd short exact sequences respectively.