The functor arrOf takes a contract for a set of elements and produces a contract for the set of lists of those elements. Concatenation of lists makes them into a monoid. Let Mon be the category of monoids and monoid homomorphisms; then we can say that arrOf is a functor from Set to Mon.

There’s another functor from Mon to Set that simply forgets the monoidal structure: it takes an element of a monoid and produces an opaque wrapper with one method, unwrap. Let’s call it setOf.

The contract setOf(arrOf(x)): Set -> Set is for a wrapped array of x‘s. Given any element xVal, we can produce a value wrap([xVal]) that passes setOf(arrOf(x)).

The contract arrOf(setOf(m)): Mon -> Mon is for lists of wrapped elements of a monoid. Given any list like

[wrap(mVal1), wrap(mVal2), wrap(mVal3)]

passing this contract, we can get a monoid element passing m by unwrapping each element and multiplying them all together.

The categories Set and Mon are not equivalent: setOf ○ arrOf is not isomorphic to the identity since there’s no way to take an element like wrap([xVal1, xVal2, xVal]) that passes setOf(arrOf(x)) and produce an element passing the contract x that doesn’t depend on x.

However, setOf and arrOf are related in a special way: with a little thought, it’s easy to see that the set of monoid homomorphisms from arrOf(x) to y is isomorphic to the set of functions passing hom(x, setOf(y)): a monoid homomorphism from arrOf(x) is completely specified by saying what it does to each of the values that pass x: you just map that function over the list and then multiply the results together.

Two functors $L:C \to D$ and $R:D \to C$ are adjoint functors if for all objects c in C and d in D,

$\hom_D(L(c), d) \cong \hom_C(c, R(d)).$

In the example above, $C = \mbox{Set},$ $\hom_C(c, c')$ is the set of functions from the set c to the set c’, $D = \mbox{Mon},$ and $\hom_D(d,d')$ is the set of monoid homomorphisms from the monoid d to the monoid d’.

Other examples of adjoint functors include

1. $\hom_{\mbox{PSet}}(L(c), d) \cong \hom_{\mbox{Set}}(c, R(d)).$ PSet is the category of “pointed sets”—i.e. sets with a special chosen element called the “point”—and functions between them that map the point of the domain to the point of the codomain. Here, $L(c)$ is the set c plus an extra point, while $R(d)$ forgets which element of d was the special one.
2. $\hom_{\mathbb{N}}(2m, n) \cong \hom_{\mathbb{N}}(m, \lfloor\frac{n}{2}\rfloor),$ where $\mathbb{N}$ is the category whose objects are natural numbers and there’s a morphism from $m$ to $n$ if $m\le n.$ Here, the floor function forgets the fractional part of $\frac{n}{2}.$
3. $\hom_{n+1}(\int_0^x f(y) dy, g(x)) \cong \hom_{n}(f(x), \frac{d}{dx} g(x)),$ where $n$ is the category of polynomials of degree $n$ and there’s a morphism from f to f’ if f is bounded above by f’. Here, the derivative operator forgets the constant term.
4. In general, any “forgetful” functor R that forgets some structure has a left adjoint L, the “free” structure of that kind.
5. $\hom_{\mbox{Set}}(c+c', d) \cong \hom_{\mbox{Set}^2}((c,c'), \Delta d),$ where Set² is the category of pairs of sets and pairs of functions between them, and $\Delta d = (d, d).$ This is a categorification of the fact that in real numbers, $d^{c+c'} = d^c d^{c'}.$
6. $\hom_{\mbox{Set}^2}(\Delta c, (d, d')) \cong \hom_{\mbox{Set}}(c, d \times d').$ This is a categorification of the fact that in real numbers, $(dd')^c = d^c d'^c.$
7. $\hom_{\mbox{Set}}(c \times c', d) \cong \hom_{\mbox{Set}}(c, \hom_{\mbox{Set}}(c', d)).$ This is a categorification of the fact that in real numbers, $d^{cc'} = (d^c)^{c'}.$
8. $\hom_{\mbox{Sub}}(c \cup c', d) \cong \hom_{\mbox{Sub}^2}((c,c'), \Delta d),$ where Sub is the category of sets and inclusions (i.e. there’s a morphism $c\to c'$ if $c \subset c'.$)
9. $\hom_{\mbox{Sub}^2}(\Delta c, (d, d')) \cong \hom_{\mbox{Sub}}(c, d \cap d').$
10. $\hom_{\mbox{Bool}}(\exists x \in c \mbox{ such that } x \in s, d) \cong \hom_{\mathcal{P}s}(c, K(d)).$ Here Bool is the category with two objects F and T and an arrow from F to T—that is, there’s an arrow from one object to another if the one implies the other. $\mathcal{P}s$ is the set of subsets of $s,$ and $K(d)$ maps F to the empty set and T to $s.$
11. $\hom_{\mathcal{P}s}(K(c), d) \cong \hom_{\mbox{Bool}}(c, \forall x \in d, x \in s).$

Adjoint functors turn up all over the place.