In this paper: Matching for teams by G. Carlier and I. Ekeland (2010), https://www.jstor.org/stable/25619994?seq=1#metadata_info_tab_contents: they claim that follows a standard argument in Ekeland and Teman (1999), we rewrite
$$\inf_{\nu\in\mathcal{M}(Z)} \sum_{j=0}^N F_j^*(\nu)=-(\sum_{j=0}^N F_j^*)^*(0) \qquad (*)$$ where $F^*$ is the Fenchel transform on page 407.
Here for $\varphi\in C^0(Z,\mathbb{R})$, define $$F_j(\varphi):=-\int_{X_j} \varphi_j^{c_j}d\mu_j$$ where $\varphi_{j^{c_j}}=inf_{z\in Z}\{c(x,z)-\varphi(z)\}$.
I am confused about using which argument to get this identity $(*)$. Why can we have
$$\inf F_j^*(\nu) =-(F_j^*)^*(0) ?$$
I guess we need some arguments in convex analysis.