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TL;DR: I want to check whether a forgetful functor $F: POSET \to SET$ that 'forgets' orders i.e. $F((X, \le)) = X, F(f: (X, \le) \to (X', \le')) = f: X \to X'$ has left and right adjoint.

Let $Set$ be category of sets and functions between them as morphisms. Let $POSET$ be category of partial orders and monotone morphisms between them that preserve orders.

Left adjoint is a functor $G: Set \to POSET$ such that for all objects $A \in Set, G(A) \in POSET$ is free over $A$. Right adjoint is the same, but with cofree objects.

Then there is adjunction functor theorem that states if $K'$ is locally small complete category, then a functor $F: K' \to K$ has a left adjoint iff $F$ is continous and for each $A \in K$ there exists a set $\{ f_i: A \to F(X_i), i \in I\}$ (of objects $X_i \in K'$ with morhisms $f_i: A \to F(X_i)$) such that for each $B \in K' and h: A \to F(B)$ for some $f: X_i \to B, i \in I$ we have $h = f \circ f_i$. But I don't understand the second requirement.

From what I understand the left adjoint of $F$ would be a functor $G: Set \to POSET$ that would map sets to discrete posets, but I am unsure about this.

Nodreh
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    One half of your question (the part about having a right adjoint): https://math.stackexchange.com/questions/3507361/forgetful-functor-from-pos-to-set-does-not-have-a-right-adjoint/3507384#3507384 – Mark Kamsma Feb 20 '23 at 14:37

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The adjunction theorem is cool, but I have never used it in practice. Whenever an adjoint exist, it is usually not to hard to construct it explicitly. This is also true in this case:

The comment by Mark Kamsma and the linked question show that the forgetful functor $U$ does not preserve colimits. Hence it can not have a right adjoint. It has a left adjoint though. Given a set $S$, let $FS$ be the poset with underlying set $UFS=S$ and give it the corsest possible relation $s\leq s$ for each $s\in S$ and nothing more. You can then easily check that $Poset(FS,P)=Set(S,UP)$ for each poset $P$. The counit is the identity map $UFS=S$, and the $P$th component $\eta_P$ of the unit satisfies $U\eta_P= id_{UP}$.

If you are more interested into the adjunction existence theorem in general than into the concrete construction, write me a comment.

Nico
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