c-concavity and c-transform

Question

I am having trouble in understanding $c$-concavity and the $c$-transform from Villani's book on optimal transport. Let $c:X\times X\to\mathbb{R}\cup\{\infty\}$ be a cost function and $\psi:X\to\mathbb{R}\cup\{\infty\}$. The $c$ transform $\psi $ is defined as $$\psi^c(y) = \inf\limits_{x\in X}(\psi(x)+c(x,y))$$ which is a generalisation of the Legendre transform when taking a cost $c(x,y)=-\langle x,y\rangle$. Indeed, with this cost we get $$\psi^c(x)=\sup\limits_{x\in X} (\langle x,y\rangle-\psi(x)).$$ But now a function $\phi$ is said to be $c$-concave if there exists $\psi$ such that $\phi=\psi^c$. But if we take again the cost $c(x,y)=-\langle x,y\rangle$ this means that $\phi$ is the Legendre transform of $\psi$, so $\phi$ should be convex not concave no?