surjection modulo jacobson radical for a finitely generated module

Let be a module-homomorphism between -modules, an ideal contained in the Jacobson radical.
Suppose that is finitely generated and that is surjective.

Then so is .

Recall that a module homomorphism is surjective if and only if the cokernel vanishes.
Observe that is also finitely generated, as it is the quotient of a finitely generated module.
Using the counit of the tensor hom adjunction gives a commutative diagram

Now is right exact, hence we may identify , where the latter vanishes as is surjective by assumpiton.

This shows that is surjective or equivalently .
Hence Nakayama Lemma for Jacobson Radical shows that as is finitely generated as quotient .
This shows that is surjective.