The isolation of the notion of an adjunction is one of the most important contributions of category theory. In a sense adjoints form the first ‘non-trivial’ part of category theory; at least it can seem that way now that all the basic stuff has been sorted out. There are adjunctions all over mathematics, and examples were known before the categorical notion was formalized. We have already met several examples, and later I will point you to them.

In this chapter we go through the various aspects of adjunctions quite slowly. We look at each part in some detail but, I hope, not in so much detail that we lose the big picture.

There is a lot going on in adjunctions, and you will probably get confused more than once. You might get things mixed up, forget which way an arrow is supposed to go, not be able to spell contafurious, and so on. Don't worry. I've been at it for over 40 years and I still can't remember some of the details. In fact, I don't try to. You should get yourself to the position where you can recognize that perhaps there is an adjunction somewhere around, but you may not be quite sure where. You can then look up the details. If you ever have to use adjunctions every day, then the details will become second nature to you.