I heated pasteurized milk to around 80 degrees Celsius and let it cool down to 45 degrees Celsius (according to a few guides) before adding 50 mL vinegar. But I've been wondering, why is it considered the optimal temperature for coagulating milk? (Note: This was for an experiment and I don't intend to add any starter cultures. I'm just investigating the effect of acids on the coagulation of cow milk.)
45C is not the optimal temperature for the acid coagulation of milk. There are a variety of methods for milk coagulation. Different variables are at play in each system. For acid coagulation, milk is heated to 80C for at least 5 minutes to denature the proteins. The addition of acid, and continued heating causes the coagulation. Perhaps you are confusing the necessity of cooling with adding live cultures, such as in yogurt making. In this case, the cooling step is necessary to provide an environment that supportive for the growth of the live cultures.