Lecture
This lecture concludes the proof of the existence of left derived functors by constructing the required natural transformation and establishing its universal property. The special case where the functor is a composite of a functor between model categories, post-composed with the localization functor to the homotopy category, is also discussed, leading to the concepts of total left and right derived functors. The lecture covers the commutativity of diagrams, the construction of natural transformations, and the properties of total left and right derived functors.