The adjoint triple $\text{Free}⊣𝑈⊣\text{Cofree}$ induces a monad $𝑇=𝑈\circ \text{Free}$ and a comonad $𝑊=𝑈\circ \text{Cofree}$ on $\text{Fam}(\mathrm{𝒞︀})$.

Both comparison functors are equivalences: presheaves are the Eilenberg–Moore algebras of $𝑇$ and the co-Eilenberg–Moore coalgebras of $𝑊$,

So presheaves are both monadic and comonadic over families.

Reading the algebra structure concretely: a $𝑇$-algebra on a family $𝐴$ is a map ${\sum }_{𝑦}\mathrm{𝒞︀}(𝑥,𝑦)\times 𝐴(𝑦)\to 𝐴(𝑥)$ for each $𝑥$, subject to the monad algebra laws — that is, exactly a functorial action of restriction.

The comonadic reading is the same structure seen from the element’s side: a $𝑊$-coalgebra is a map $𝐴(𝑥)\to {\prod }_{𝑦}\mathrm{𝒞︀}(𝑦,𝑥)\to 𝐴(𝑦)$, giving each value its restriction along every morphism into $𝑥$. Where the monad says restriction acts on values, the comonad says a value already carries all of its restrictions — and the coalgebra laws say it does so coherently.

Fix a bicategory $\mathrm{𝒦︀}$, with composition $\star$, identity 1-cells $1_{𝑥}$, associator $𝛼$, and unitors $𝜆,𝜌$. A monad in $\mathrm{𝒦︀}$ internalises the usual notion of monad: it is an endo-1-cell carrying a multiplication and a unit that satisfy the monoid laws up to the coherence cells of the bicategory.

A monad in $\mathrm{𝒦︀}$ consists of

1. a 0-cell $𝑥$, the object the monad acts on;
2. an endo-1-cell $𝑡:\mathrm{𝒦︀}(𝑥,𝑥)$;
3. a multiplication 2-cell $𝜇:𝑡\star 𝑡\Rightarrow 𝑡$;
4. a unit 2-cell $𝜂:1_{𝑥}\Rightarrow 𝑡$;

5. such that $𝜇$ is associative: the following diagram of 2-cells commutes in $\mathrm{𝒦︀}(𝑥,𝑥)$, where the top map is the associator that rebrackets the threefold composite:

6. and such that $𝜇$ and $𝜂$ satisfy the unit laws: the following two diagrams commute in $\mathrm{𝒦︀}(𝑥,𝑥)$, where the hypotenuses are the left and right unitors:

Taking $\mathrm{𝒦︀}$ to be the bicategory of categories, functors, and natural transformations recovers an ordinary monad on a category: $𝑡$ is the endofunctor, $𝜇$ the multiplication, and $𝜂$ the unit, with the coherence cells all identities.