Consider the functional equation

$$ g\left(a\right) = \int_0^1 \frac{e^{h+f(h)}}{1+e^{h+f(h)}}dh $$

and this holds for all $a$. $g(a)$ is a known function on a continuous support. How can I show that there is one unique $f(h)$ that solves this equation?