Given the prox operator i.e.

$$ \operatorname{prox}_{ h \left( \cdot \right) } \left( x \right) = \arg \min_{u} h \left( u \right) + \frac{1}{2} {\left\| u - x \right\|}_{2}^{2} $$

the Moreau decomposition property says that

$$ x = \operatorname{prox}_{ h \left( \cdot \right) } \left( x \right) + \operatorname{prox}_{ {h}^{\ast} \left( \cdot \right) } \left( x \right) $$

where $h^*$ is the conjugate of $h$

I was reading a proof of this which went as follows :

Define $ u = \operatorname{prox}_h (x)$ and $v = x - u$

From optimality condition of minimization in the definition of the prox operator, $ x-u \in \partial h(u)$, so $ v \in \partial h(u)$

- $u=x-v \in \partial h^* (v)$, hence $v = \operatorname{prox}_{h^*} (x) $

I didn't understand the 3rd step of the proof, i.e. how $ u \in \partial h^* (v)$ follows from $ v \in \partial h(u)$. Could someone shed some light on this?