The Inverse of the Link Function is the Gradient of the Convex Conjugate

Observe that the gradient of the dual is the inverse of the gradient of the log-partition,

\[\begin{aligned} f(g(\theta)) &= f(\mu) \\ &= \nabla_\mu F(\mu) \\ &= \nabla_\mu \Big[ \mu \theta - G(\theta)\Big] \\ &= \theta + \mu \nabla_\mu \theta - g(\theta) \nabla_\mu \theta \\ &= \theta. \\ \end{aligned}\]

The converse is similar, so $f = g^{-1}$.