A filtered poset and a filtered diagram (category)

Question

Let $X$ be a poset.

Statement: A subset $Y\subseteq X$ is filtered if an only if there exists a filtered diagram (category) $D$ with a functor $D\rightarrow X$ such that the image of $D$ is $Y$.

How to prove this?

Can it be generalized?

Answer 1

The "only if" direction is easy: take $D = Y$ and the inclusion. For the "if" direction, just proceed directly: take two elements in $Y$; then they have a preimage in $D$, so they have an upper bound in $D$, so they have an upper bound in $Y$, etc.

Of course, one should note that filtered diagrams in the sense of category theory are upside down compared to filters in the sense of order theory.