A family over $\mathrm{𝒞︀}$ is a set $𝐴(𝑥)$ for each object $𝑥$, with no action of morphisms. Families form a category $\text{Fam}(\mathrm{𝒞︀})$: a morphism $𝐴\to 𝐵$ is a function $𝐴(𝑥)\to 𝐵(𝑥)$ for each $𝑥$.

Forgetting the restriction maps of a presheaf gives a functor

It has both a left and a right adjoint,

The two adjoints demonstrate different means of forcing a family to be functorial. The right adjoint universally quantifies over morphisms in,

with restriction along $𝑓$ given by precomposition. The left adjoint instead existentially quantifiers over morphisms out:

with restriction acting on the first component. (For $\text{Free}$ we ask that $\mathrm{𝒞︀}$ have a set of objects, so that this sum is a set and thus $\text{Free}$ defines a presheaf.)