I am watching a course about RC circuits and it got to this circuit. I understand how it functions but I have a problem understanding what sign each voltage should have. After writing the KVL for the circuit I came up with the answer in the image. But the professor who is teaching the course got the answer in the second image. Is my KVL wrong?

 

When you write Vc1 you use I1 going out of the capacitor, you use it as positive.
When you write VC2 you use I1 going into the capacitor, and you still use it as positive.
That's where the flipping sign comes from.

 

I understand now, thank you.