These are the columns you are looking for.
Lean (of peak) is cooler too. Not looking to start the "OMFG U RUN LEAN OF PEAK OVERSQUARE CALLING FLIGHT STANDARDS" conversation here, just pointing out that leaner can be cooler.
As you pull the mixture control back, CHT increases, BHP increases, and EGT increases. Brake horsepower specific fuel consumption decreases (lower is better). Power (BHP) peaks first, followed by CHT, then EGT. At 50F rich of peak EGT, a common cruise mixture setting, CHT is actually highest (lower is better). At peak EGT, BHP and CHT are lower than at 50F rich of peak. On the lean side of peak EGT, BHP and CHT are lower than at 50F rich of peak.
To be precise and nitpicky: very high power settings require a richer than normal mixture to retard the combustion event and prevent detonation. Retarding the combustion events results in a lower total internal cylinder pressure, which means that temperatures will be lower. The cooling is a byproduct of the former.