Generally I would be opposed to a city paying for 20% of a private development's costs, however, the city forced its own hand in this case by not spending enough money downtown to begin with (neglect), not spending it wisely (neglect), and other consistent poor decisions that have left Jacksonville in a ditch so to speak such that natural momentum isn't even there as a force, economically, over the whole region.
For a sunbelt FL city, growth is slow now. Job growth is slow and not meaningful or conducive to downtown or city development. Downtown is long neglected. We're at a point where frankly the city needs to full on subsidize projects such as this. A healthy downtown will do wonders for the whole city in the future, and could certainly impact population growth and job growth going forward as it's a "sign" of the health and attractiveness of the city. It should spend the money.