Is there a mapping reduction for every two language $A$ and $B$ to some language $C$?
The definition of the reduction is wrong.
We actually need to define two reductions: from $A$ to $C$ and from $B$ to $C$. The reduction from $A$ to $C$ maps $w$ to $0w$. The reduction from $B$ to $C$ maps $w$ to $1w$.