Is there a $1$ to $1$ correspondence between probability semigroups and generators?

Question

I am currently reading Liggett's Continuous Time Markov Processes. Fix an appropriate state space $E$. Let $A$ be the set of all probability semigroups and $B$ the set of all generators. Theorem $3.16$ gives us a function $f$ from $A$ to $B$. Theorem $3.24$ then defines a function $g$ from $B$ to $A$ and tells us that $f\circ g$ is the identity on $B$. Do we also know that $g\circ f$ is the identity on $A$ (and hence $g=f^{-1}$)?