Yeah, basically yes.Do you mean the physics behind the connection between carbon dioxide and temperature? If that's the case I wrote something below.
The role of carbon dioxide (and other greenhouse gases, such as methane and water vapor) in regulating the temperature of the planet is relatively straightforward. I'll divide this into feedback 1: the direct effect of increases in CO2, and feedback 2: the indirect effect.
Feedback 1. Carbon dioxide concentration is very little by mass but has strong radiative effects. This radiative effect occurs because carbon dioxide preferentially absorbs radiation in the infrared, which is then re-emited. So, basically solar radiation, with most of the spectrum being in the visible range, gets to the ground relatively unimpeded. This heats the earth, which in turns radiates back to space, with the bulk of the radiation in the infrared*. Since carbon dioxide is efficient at absorbing at the infrared, a good portion of the energy radiated by the earth is trapped in the atmosphere, which then heats it and radiates back a portion to the earth heating it more. This is basically the greenhouse effect, and is key to human survival. Without this effect the earth temperature would be approximately 33C lower. Now, the problem is that with higher man-made CO2 concentrations this effect occurs a little bit too much than before, so the earth equilibrium temperature rises. Now, by itself, increases in CO2 would yield about a quarter of the heating that what models project for end of the century, and the rest is given by the indirect effect (feedback 2).
Feedback 2. CO2 increases give the first kick to increases in temperature. When temperature increases the amount of water vapor also increases**. Water vapor is actually a stronger greenhouse gas compared to carbon dioxide. So, taking that into account, increases in water vapor (due to increases in CO2 and other greenhouse gases) is the main driver of already observed and future increases in temperature.
*it turns out there is a temperature - peak of radiation spectrum relationship, higher temperature objects preferentially radiates at the higher energy end: visible, ultraviolet, so forth.
**close to exponentially, in what is known as the Clausius-Clapeyron relation (Clausius–Clapeyron relation - Wikipedia). Hot places can have quite a bit more of water vapor before it condenses. Same amount of water vapor in a colder place would just form water droplets and eventually rain.
And here is my argumentation (please correct me when I'm wrong). Given the strong (to say the least) correlation between CO2 and temperature, it is safe to say that they have to correlate in some way. As you described in Feedback 1, CO2 influences the rise of the temperature via greenhouse effect. The other way around the majority of the CO2 is solved in the oceans (50 times as much CO2 solved in oceans compared to the atmosphere) and the solubility decreases with increasing temperature. I think both of these effects are scientifically proven and I don't think there's any doubt or controversy about their existence.
The change in temperature in this timespan is a cyclic one (till excessive human activity kicked in), induced by the Milankovic cycles (undoubtably influencing the temperature primarily). And here's my probably "unique" take on this: While there is certainly an amplification in the increase of the temperature due to the emitted CO2 (and other effects like feedback 2) because of the Milankovic-induced rise in temperature, I suggest that only one way of the causality can be a dominant one, because otherwise we would end up with some sort of a positive feedback amplification instead of the cyclic process we are dealing with here. So for the 400,000 years displayed here, the changing temperature is setting the impulse for the change in atmospheric CO2 concentration, while for the last 250 years CO2 is supposed to be the main initiator of the change in temperature.
I know overall it is probably more complicated than that, but when I check sites that call themselves something like climatefacts or so, they often base their arguments on the fact that the current rise in CO2 occurs before the rise of the temperature while before it happened to be vice-versa or end their arguments with something like "the current change is too severe to be caused by non-humane factors". While I consider the latter "argument" as rather pointless when the person reading isn't even denying the human influence on the climate, the former can also easily be explained by the (in that case additional) post-industrial emission of CO2 by humans which occurs rather independantly from the rise in temperature (while the overall CO2 level would obviously still be influenced by the rising temperature additionally). The warming would then primarily be caused by one of the other endless severe human encroachments, that are obviously all correlating with both, temperature and CO2 level.
We would then basically have two "types" of CO2 in the atmosphere, the humanly emitted CO2 (that makes the CO2 curve rise earlier then the temperature one) and the "normal" temperature-caused CO2.