Say I have two sets of values $A$ and $B$ and for each set I have a computable function from that set to a third set $C$. Now suppose that I want to construct a function from $A$ to $B$, such that if I compose that function with the $B$ to $C$ function mentioned above I get a function that produces the same results as the $A$ to $C$ function mentioned above.
If I know the time-complexity of the two functions that return elements of $C$, does that allow me to say anything about a function from $A$ to $B$ with the specified property? For example can any bounds be placed on the computational complexity of such a function? Can we even say whether such a function is computable or not?