Can a MacBook Pro 13 (Mid-2010) battery be used in a 2009 unibody polycarbonate MacBook?
No, the batteries used in these respective models are not compatible with each other.

If it helps, the battery for the Mid-2010 13" MacBook Pro is compatible with a number of 13" MacBook Pro models from Mid-2009 until Mid-2012.

As for the battery in the Late-2009 13" polycarbonate MacBook, this is compatible with the Mid-2010 13" MacBook.