John, in [Comment 25](https://forum.azimuthproject.org/discussion/comment/16703/#Comment_16703), 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?