TITLE: Posets isomorphic to their endomorphism poset
QUESTION [15 upvotes]: Let $(P,\leq)$ be a poset. We set $$\text{End}(P)=\{f: P\to P: f\text{ is order-preserving}\}$$ and order $\text{End}(P)$ pointwise.
Is there a poset with more than 1 point such that $P\cong \text{End}(P)$?

REPLY [5 votes]: The answer to 

Is there a poset with more than 1 point such that $P\cong\text{End}(P)$?

is No. This follows immediately from Theorem 3 in

Roy O. Davies, Allan Hayes and George Rousseau, Complete lattices and the generalized Cantor theorem, Proc. Amer. Math. Soc. 27 (1971), 253–258.

This reference was pointed out to me by user bof in a (now deleted) comment to this post. More precisely bof gave a link to this answer of them.