Consider the functional equation

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

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