"If you have a good set of functional tests, is it necessary to write unit tests?".
YES, unit tests shall be applied to each module before functional tests, even before the whole system/program is ready. And should save a lot of debugging time.
As matija suggested above a good set of functional tests guarantee that the product works fine for the time being. However, consider future extentions - individual units may become exposed to inputs you have never dreamt of.
But if you are limited in your time - consider writing "release notes".
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