**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:

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