Your construction is correct. I have only one remark.
Whether this explanation is sufficient depends on the requirements of your course. To be strict, one has to have an argument by induction on the length of the word. For this, one has to generalize the statement "A word is accepted by the automaton iff it is generated by the grammar" to describe the intermediate situation, when only a part of the word has been read and generated. It's a good exercise to formulate such general induction statement; after that, the base case and the step are pretty obvious.
Note that the grammar you constructed is called a right regular grammar.