Moreau Decomposition in Terms of the Proximal Operator of the Conjugate Function

Question

By the Moreau decomposition, $x = \text{prox}_{tf}(x) + t\cdot\text{prox}_{t^{-1}f^*}(x/t)$.

If $f$ is a closed, convex function, then is it equivalent to write $x = \text{prox}_{tf^*}(x) + t\cdot\text{prox}_{t^{-1}f}(x/t)$?

This second form seems more useful for applying the decomposition to get the proximal operator of $f^*$, given $f$.

Source: L. Vandenberghe - The Proximal Mapping.