You've also got the waste heat from power generation and transmission required to power all the ACs that plug into the mains and draw their power from the grid. If an AC is powered by locally connected solar or whatever that's going to be pretty insignificant, but the rest are going to add up. The laws of thermodynamics totally apply, which means you are not magically moving heat from A to B in a zero sum game, you're consuming power to do it, and that means more waste heat in addition to the losses in the system through inefficiencies.
Cities are already microclimates and mass AC adoption is absolutely going to cause an aggregate, and almost certainly measurable, temperature increase across that microclimate, and especially so in narrow streets where there is limited airflow to disperse that extra heat. If the heatwave is already making outside temperatures unhealthy then adding another degree or whatever on top of it to help keep interiors cool via AC is going to really suck for those who are forced to go outside in it for whatever reason.
A better solution, given climate change has been a thing for decades, would have been to look at how pre-AC civilizations in equatorial regions built very efficient ways of keeping interiors cool passively in extreme heat and adopted those techniques in anticipation of hotter summers for any new builds over (realistically) the last 20-30 years. Hindsight, and naive optimism the Paris Agreement et al would work, is a wonderful thing though and here we are - putting a band aid in place that will actually make the overall problem slightly worse.