# homological algebra – Exercise with exact sequence in an abelian category

Suppose (in an abelian category) to have two exact sequences, $$0to Axrightarrow f Bxrightarrow g Cto 0$$ and $$0to A’xrightarrow {f’} B’xrightarrow {g’} C’to 0$$, and three arrows $$a:Ato A’$$, $$b:Bto B’$$ and $$c:Cto C’$$ such that the two squares that arise are commutative. I must prove that if $$a$$ is an isomorphism, then$$xleftarrow b Bxrightarrow g$$ is the pullback of $$g’$$ and $$c$$. I don’t know if it’s useful, but it should be sufficient to prove the same statement with $$A’=A$$ and $$a=1_A$$; however I don’t see how, given two maps $$h:Dto B’$$, $$k:Dto C$$ such that $$g’h=ck$$, one could define a map from $$D$$ to $$B$$.