**MD Puzzle 2**: Find the *right adjoint* \\(r : A\times A \to A\\) to \\(\Delta\\) such that:

$$
\Delta(x) \leq_{A\times A} (y,z) \Longleftrightarrow x \leq_{A} r(y,z)
$$

Directly taking John's tutorial and dropping in the functions in the appropriate places we get,

>If \\(\Delta: A \to A\times A\\) has a right adjoint \\(r : A\times A \to A\\) and \\(A\\) is a poset, this right adjoint is unique and we have a formula for it:

$$ r(x,y) = \bigvee \\{a \in A : \; \Delta(a) \leq_{A\times A} (x,y) \\} . $$


**MD Puzzle 3**: Find the *left adjoint* \\(l : A\times A \to A\\) to \\(\Delta\\) such that:

$$
l(x,y) \leq_{A} z \Longleftrightarrow (x,y) \leq_{A\times A} \Delta(z)
$$

>If \\(\Delta: A \to A\times A\\) has a left adjoint \\(l : A\times A \to A\\) and \\(A\\) is a poset, this left adjoint is unique and we have a formula for it:

$$ l(x,y) = \bigwedge \\{a \in A : \; (x,y) \leq_{A\times A} \Delta(a) \\} .$$