In this series of talks, I will prove that two standard notions of "homotopy coherent natural transformation" (one defined in terms of homotopy coherent ends, the other in terms of homotopy coherent nerves, both due to Cordier and Porter) are equivalent, in the sense that they define the objects of equivalent Kan complexes. I will use this result to give an alternative proof of a theorem of Lurie which compares two models for the infinity-category of presheaves over a simplicial set.