There's another aspect to this that hasn't been mentioned yet, but is related to **Puzzle 46**.

Suppose \$$A\$$ is a preorder. The subsets of \$$A\$$ form a poset \$$PA\$$.

There is a natural monotone function \$$A\rightarrow PA\$$ sending \$$a\in A\$$ to \$$\operatorname{\downarrow}a = \\{x\in A : x\le a\\}\$$.

Now given any \$$S\in PA\$$, \$$S\subseteq{\operatorname{\downarrow}a}\$$ iff \$$a\$$ is an upper bound of \$$S\$$.

And if the join of \$$S\$$ exists, \$$a\$$ is an upper bound of \$$S\$$ iff \$$\bigvee S \le a\$$

So if \$$A\$$ has all joins, \$$\bigvee S \le a\$$ iff \$$S\subseteq{\operatorname{\downarrow}a}\$$

ie \$$\bigvee\$$ is a left adjoint to \$$\downarrow\$$