Let $\mathrm{𝒞︀}$ carry a direct structure. The strict downset $\text{𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇}(𝑥)$ of an object $𝑥$ is the presheaf of morphisms into $𝑥$ from strictly lower objects:

with restriction by precomposition — well defined since degrees are non-decreasing, so precomposing can only stay strictly below.

The evident inclusion $\text{𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇}(𝑥)\rightarrowtail よ𝑥$ makes $\text{𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇}(𝑥)$ a sieve on $𝑥$. It is moreover a proper sieve, as it exlcudes the identity.