ok, like other diagram chasing problems the solution is fairly straightforward but quite lengthy and annoying! i'll give the solution this time and that would also be the last time!

so the rows in your commutative diagram are we also have the sequence defined by:

sosuppoose first that in your diagram is an isomorphism. see that since is injective, is injective too.

also by the definition: but since the diagram is commutative, we have thus i.e. next we'll show that :

let i.e. hence: which gives us because is an isomorphism. so

thus for some now since we'll have because is injective. hence:

so we've proved that finally we need to show that is surjective: let then because is an isomorphism. so

for some now since is surjective, for some thus: which gives us: so there exists

such that hence this completes the proof of the first half of the problem.

conversely, suppose that the sequence is exact. we want to show that is an isomorphism. first we show that is injective: so suppose for some since

is surjective, we have for some then: and hence so for some which gives us:

thus so there exists such that hence and thus

finally we need to show that is surjective: let then because is surjective. now since is surjective, there exists such

that: thus: