Welcome to the last installment in our mini-series on adjunctions in category theory. We motivated the discussion in Part 1 and walked through formal definitions in Part 2. Today I'll share some examples. In Mac Lane's well-known words, "adjoint functors arise everywhere," so this post contains only a tiny subset of examples. Even so, I hope they'll help give you an eye for adjunctions and enhance your vision to spot them elsewhere.

An adjunction, you'll recall, consists of a pair of functors $F\dashv G$ between categories $\mathsf{C}$ and $\mathsf{D}$ together with a bijection of sets, as below, for all objects $X$ in $\mathsf{C}$ and $Y$ in $\mathsf{D}$.

In Part 2, we illustrated this bijection using a free-forgetful adjunction in linear algebra as our guide. So let's put "free-forgetful adjuctions" first on today's list of examples.

Last time I shared a light introduction to adjunctions in category theory. As we saw then, an adjunction consists of a pair of opposing functors $F$ and $G$ together with natural transformations $\text{id}\to\ GF$ and $FG\to\text{id}$. We compared this to two stricter scenarios: one where the composite functors equal the identities, and one where they are naturally isomorphic to the identities. The first scenario defines an isomorphism of categories. The second defines an equivalence of categories. An adjunction is third on the list.

In the case of an adjunction, we also ask that the natural transformations—called the unit and counit—somewhat behave as inverses of each other. This explains why the ${\color{red}\text{arrows}}$ point in opposite directions. (It also explains the "co.") Except, they can't literally be inverses since they're not composable: one involves morphisms in $\mathsf{C}$ and the other involves morphisms in $\mathsf{D}$. That is, their (co)domains don't match. But we can fix this by applying $F$ and $G$ so that (a modified version of) the unit and counit can indeed be composed. This brings us to the formal definition of an adjunction.

Some time ago, I started a "What is...?" series introducing the basics of category theory:

• "What is a category?"
• "What is a functor?" Part 1 and Part 2
• "What is a natural transformation?" Part 1 and Part 2

Today, we'll add adjunctions to the list. An adjunction is a pair of functors that interact in a particularly nice way. There's more to it, of course, so I'd like to share some motivation first. And rather than squeezing the motivation, the formal definition, and some examples into a single post, it will be good to take our time: Today, the motivation. Next time, the formal definition. Afterwards, I'll share examples.

Indeed, I will make the admittedly provocative claim that adjointness is a concept of fundamental logical and mathematical importance that is not captured elsewhere in mathematics.

- Steve Awodey (in Category Theory, Oxford Logic Guides)