How to define the category Pos of posets as a presheaf The category of Sets, $\textbf{Set}$, can be described as a presheaf, namely, take the category of one object with the identity morphism, the functors from that to Sets give the different sets in $\textbf{Set}$. 
How can you describe the category $\textbf{Pos}$ of posets this way? I thought that you could take another category with one object, with the identity arrow, but also other arrows that then give the relations in the poset. Maybe an arrow can describe a relation if the functor maps it to a function in sets, that maps an element to another element that is the most closely related and otherwise just to itself? (so if $x \leq y \leq z$, then the function in Sets maps $x$ to $y$, $y$ to $z$ and the rest to itself?)
 A: This is not possible: there are properties that every presheaf category has that $\mathbf{Pos}$ does not.  For instance, in any presheaf category, any morphism that is both monic and epic is an isomorphism (proof sketch: a morphism is monic iff it is pointwise injective and epic iff it is pointwise surjective).  But this is not true in $\mathbf{Pos}$.  For instance, let $X=\{a,b\}$ with the discrete ordering ($a$ and $b$ are incomparable) and let $Y=\{a,b\}$ with the ordering $a\leq b$.  Then the identity map $X\to Y$ is order-preserving and is both monic and epic in $\mathbf{Pos}$, but is not an isomorphism (its inverse is not order-preserving).
A: No, not as an arbitrary presheaf. We're sunk pretty much from the get-go. The theory of posets consists of a binary relation symbol, i.e. a monomorphism $R\rightarrowtail X\times X$ so already we need the functors to preserve finite products and monomorphisms. Typically what are used are what are called left exact (lex) or finite limit theories (these are synonyms) or (ambiguously) cartesian theories, and we then require the functors to be left exact/finite limit preserving/cartesian(again, ambiguously).  A left exact theory is more than we need (per Kevin Carlson's comment and the linked paper, a Horn theory suffices), but we still need more structure (such as finite products) than an arbitrary functor will preserve.
