It is known that if $f:A\to B$ is a homomorphism and $g:B\to C$ is another homomorphism, then $g\circ f:A\to C$ is a homomorphism. In other words, the composition of two homomorphisms is a homomorphism.
Suppose now that $f:A\to B$ is a homomorphism and $h:A\to C$ is a homomorphism. Is there any theorem proving that the map $g:B\to C$ is a homomorphism if $g\circ f=h$?
This is wrong.
The moral reason for this is that the condition $g\circ f=h$ tells you what $g$ does on the range of $f$ only. It could do literally anything out of the range of $f$, this condition would still be satisfied but the homomorphism condition would stand no chance to hold, since it requires much discipline.
On the other hand, it is true if $f$ is surjective.