Longer-term

Over millennia, CO2 rise has historically lagged behind temperature by centuries (150 to 1,000 years). Ice core data shows this and again indicates the causality relationship is as temperature goes up, CO2 follows.

This is further discussed in the books 'Natural and Anthropogenic Climate Variability' from which the following graphic is taken

There are several mechanisms that explain why CO2 rise follows temperature rise.

The seasonal increase in CO2 may also have a longer-term eleement.

Warm oceans dissolve less CO2 than cold oceans and since the mass of CO2 dissolved in oceans is about 50 times greater than in the atmosphere this is a highly probable mechanism.