Earth system models (ESMs) are the principal tools used in climate science to generate future climate projections under various atmospheric emissions scenarios on a global or regional scale. Generative deep learning approaches are suitable for emulating these tools due to their computational efficiency and ability, once trained, to generate realizations in a fraction of the time required by ESMs. We extend previous work that used a generative probabilistic diffusion model to emulate ESMs by targeting the joint emulation of multiple variables, temperature and precipitation, by a single diffusion model. Joint generation of multiple variables is critical to generate realistic samples of phenomena resulting from the interplay of multiple variables. The diffusion model emulator takes in the monthly mean-map of temperature and precipitation and produces the daily values of each of these variables that exhibit statistical properties similar to those generated by ESMs. Our results show the outputs from our extended model closely resemble those from ESMs on various climate metrics including dry spells and hot streaks, and that the joint distribution of temperature and precipitation in our sample closely matches those of ESMs.