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@BenjaminSteinberg I don't think so. For example, say $S = \{ x,y,z\}$ with $x< z$ and $y< z$. Then in the extended partial order, it can happen that $xy< z$ or not.
@BenjaminSteinberg It's like I've found a new gadget, and I'm looking for the instruction book. I don't specifically have in mind that left adjoint, but I'd be happy to read about it.
