Carbon Pricing -- Wait Just One Damn Minute

OK, so the theory behind carbon pricing is that the consumption of carbon is priced artificially low, because there is no way to apply the communal costs of that pollution back to the consumer, right? So the whole point of carbon pricing is to apply a price to that externality, causing the cost of consuming carbon to go up. This makes other, non-carbon-consuming uses more economical, and it collects money on the theory that today's consumption causes damage to tomorrow.

So my question: how can you justify spending the money you are collecting as a tax on tomorrows damages on anything other than those damages, at any other time than this nebulous tomorrow? Otherwise you'll end up in the same boat in the future -- with a damaged tomorrow with no revenues extracted from the past to cover it. In essence, it double-taxes the damages on tomorrow, since future governments will still have to raise revenues to deal with said damages.

I'm not very clear this morning, there's probably a way to put this better.