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