uniqueness of chain map for direct sum of

Question

I'm trying to show that for the direct sum of two chain complexes $A \bigoplus B$ if we have two chain maps $f_A:A\rightarrow C$ and $f_B:B\rightarrow C$, there should exist a unique chain map $f:A \bigoplus B \rightarrow D$ with the property that $f_A=f \circ \iota_A$ and $f_B=f \circ \iota_B$. The maps $\iota_A,\iota_B$ are the inclusion maps of $A$ and $B$ into $A \bigoplus B$.

I believe we should define $f(a,b)=f_A(a)+f_B(b)$, but I'm not sure why this is unique.

Answer 1

It's implicit that the unique map should be a map "of the same type." In this case, that type is a homomorphism of complexes of abelian groups.