$\begin{array}{cc}{h}^{*}(\mathcal{F})& ={\sum }_{i=1}^{N}h(f_i)\hfill \\ & ={\sum }_{i=1}^{N}h(f_i^{-1})\hfill \\ & =h^{*}({\mathcal{F}}^{-1}).\hfill \end{array}$

Theorem 7.3 of [7]