Re "joins *are* left adjoints, meets *are* right adjoints"...

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\\)

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\\)