The most contentious is the amount of cooling reduction caused
by backradiation.
Whether or not
increased backradiation from an increased CO2 concentration causes the heat content (of a body of water) to increase, is determined by how much of that «energy in» flux gets converted into energy out flux in form of evaporation, etc..
The net flow of radiant heat is still upwards from the surface to the atmosphere, because the upwards thermal emission is greater than the downwards
atmospheric backradiation.
my debut SkS article, an examination of the work of William Charles Wells, who
observed backradiation from cloud in — wait for it!
Most of the heat entering the ocean arrives
via backradiation, as with land, though the microphysical mechanism is somewhat different.