The representable $\text{𝗒}𝑥$ is itself a sieve on $𝑥$. A sieve on $𝑥$ is proper when it is not equal to the representable.

Say that a proper sieve is maximal when it contains all other proper sieves as a sub-sieve.

A sieve on an object $𝑥$ of $\mathrm{𝒞︀}$ is a subobject of the representable presheaf $\text{𝗒}𝑥$: a presheaf $𝑆$ with a monic morphism $𝑆\rightarrowtail \text{𝗒}𝑥$. Sieves are a generalization from the notion of ideal found in ring theory to category theory.

A morphism $𝑓:𝑦\to 𝑥$ belongs to $𝑆$, written $𝑆\ni 𝑓$, when $𝑓$ lies in the image of the inclusion at $𝑦$. Because $𝑆$ is a presheaf and the inclusion is natural, membership is closed under precomposition:

A sieve is thus a “downward closed” collection of morphisms into $𝑥$.

Sieves on $𝑥$ are ordered by refinement: $𝑆\subseteq 𝑇$ when every morphism belonging to $𝑆$ belongs to $𝑇$.

Call a direct structure reflecting when every morphism between objects of equal degree is a split epimorphism. In a reflecting direct category, every non-invertible-in-degree morphism strictly raises degree, and the strict downset is as large as a proper sieve can be:

If the direct structure is reflecting, then every proper sieve $𝑆$ on $𝑥$ refines into the strict downset:

Suppose $𝑓\in 𝑆$ with $𝑓:𝑦\to 𝑥$ of equal degree. By reflection $𝑓$ has a section $𝑠$, and closure under precomposition gives $𝑠\star 𝑓={\text{𝗂𝖽}}_{𝑥}\in 𝑆$, contradicting properness.

So every morphism in $𝑆$ strictly raises degree. That is, every morphism in $𝑆$ is also a member of $\text{𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇}(𝑥)$.

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.