The exercise:
...Express the conjugate of $g(x) = \inf_z \{ h(z)\mid Az+b=x\}$, where $h$ is convex, in terms of $h^*$, $A$, and $b$.
In the solution manual here, they define $f$ and write
$$f^*(y,v)=\inf (y^Tx-v^Tz-f(x,z))=....$$ However, by the definition of the conjugate, $f^*(y,v)=\sup (y^Tx+v^Tz-f(x,z)).$ So, why this equality holds?