Driver's side vents blow hot air and passenger side vents blow cold air after having a new battery installed?

This can happen when the HVAC systems loses its calibration after the battery is disconnected. You can try leaving the battery disconnected overnight or try removing the HVAC fuse which is located in the fuse panel inside the car for 5 minutes.

This guide can help as well:

https://www.2carpros.com/articles/step-by-step-guide-to-reset-your-cars-hvac-system-an-essential-for-every

Here is the fuse location to help us fix the problem. Check out the images (below). Please let us know what happens.