Forcing a functor to map to a given set

Question

Let $(X, \leq)$ be a poset and $J$ a small category. Let $S$ be a subset of $X$. Viewing $(X, \leq)$ as a category, does there exists a functor $F: J \rightarrow (X, \leq)$ such that $\{F(j)\}_{j \in \textrm{ob}(J)} = S$?

Answer 1

I suppose this depends on what $J$ and $S$ are, for example if $\left|\operatorname{ob} J\right| < |S|$ then this is clearly impossible since $F$ can't be surjective on objects.

But even if $\left|\operatorname{ob} J\right| \ge |S|$ it is unreasonable to assume such a functor exists, in general.

For example, suppose $X$ has the discrete ordering ($x \le y \Rightarrow x=y$) and $J = (\mathbb{Z}, \le)$. A functor $J \to X$ is an order-preserving map of posets, so must be constant since $X$ is discrete, meaning that $\{F(j) : j \in \operatorname{ob} J\} = \{ x \}$ for some $x \in X$. So in this case no such $F$ exists for any $S \subseteq X$ of size $\ge 2$.

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Updated: To answer your more specific question from the comments: suppose $S \subseteq X$, let $J = (S, \le)$ and let $F = \iota : S \hookrightarrow X$ be the inclusion functor. Then $\operatorname{colim} F$, if it exists, is equal to $\sup S$, as you can check from the definitions.