The purpose of this exposition is threefold. Firstly, we wish to show that many concepts and arguments carry over from pre-ordered sets to categories. Secondly, we wish to make some propaganda for the notion of "subequalizer" of two functors, which appears to be more fundamental than Lawvere's so-called "commacategory", in the same sense in which equalizers are more fundamental than pullbacks. Thirdly, we wish to give simple proofs of the adjoint functor theorem and related theorems, which appear to be more economic than those in the literature. The author wishes to take this opportunity to refine some arguments that he has published earlier. He is indebted to Michael Barr, whose presentation of similar proofs in his course has provided the stimulation for preparing this note for publication, to John Isbell for his critical reading of the manuscript and to William Schelter for suggesting a shortcut in one of the proofs.