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.