Conjugation with the adjunction between presheaves and families $𝑈⊣\text{Cofree}$ lets us induce a later construction on families from the one on presheaves,

Concretely, later on families evaluates to

The later modality for presheaves on a direct category is given by the presheaf of natural transformations

out of the strict downset.

An element of $⊳𝑃$ at $𝑥$ is a coherent choice of $𝑃$-elements at all objects strictly smaller than $𝑥$.

Restriction in $⊳𝑃$ along $𝑓:𝑦\to 𝑥$ precomposes with the induced map $\text{𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇}(𝑦)\to \text{𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇}(𝑥)$. At an object of minimal degree the strict downset is empty, so $⊳𝑃$ is trivial there.

Via functoriality, every presheaf restricts to smaller indices. Thus we may define the map

that sends an element $𝑝$ over $𝑥$ to the family of all its restrictions along morphisms from strictly lower objects.