John, in [Comment 25](, there is something that confuses me:

> Here's an extreme example: for any poset \\(A\\) whatsoever, the identity function \\(1_A : A \to A\\) has a left and right adjoint, namely itself. This is easy to check straight from the definition:

> $$ a \le a \textrm{ if and only if } a \le a . $$

Shouldn't this be \\(a \le b \textrm{ if and only if } a \le b \\), no? Or why should this only hold for \\(a\\) on both sides of the comparison?