The energy that we put into the carbon dioxide in these examples will first pass through water.
People might already be familiar with the idea that you can make a fuel from water, which is a partial truth. (but one so often gotten wrong that it is one motivation for these messages).
The real trick to making a fuel from water is to put energy into breaking up the water. Having the water isn't enough.
Most of the time the energy to break apart water comes from burning fossil fuel, producing carbon dioxide, so it's a net loss to make fuel from water this way.