Find a function f : B → C such that h = f ∘ g

Question

Find a function $f : B → C$ such that $h = f ∘ g$, or prove that such function do not exist

A={1,2,3}, B={4,5,6}, C={7,8,9}

$g : A → B$
$h : A → C $
$f : B → C$

$g(1)=5, g(2)=5, g(3)=6, h(1)=7, h(2)=8, h(3)=9$

Answer 1

Hint: $h$ is bijective, but $g$ is not. In fact $|\text{im}(g)|=2$. How many elements are there maximally in $\text{im}(f\circ g)$?