Let $f$ be an injective function from $C$ to a partition $P$ of $B$, i.e. $f:C\rightarrow P$. Let $g$ be an injective function from $B$ to a partition $Q$ of $A$, i.e. $g:B\rightarrow Q$. How do I show that there is an injective function $h$ from $C$ to $Q$, i.e. $h:C\rightarrow Q$?
If the Axiom of Choice is assumed, then there is an injective function $f’$ from $P$ to $B$. Then, we can prove that $h$ exists because functions $f,f’,g$ can be composed together. How do I prove this without the Axiom of Choice?