I want to prove that the category of chain complexes of R-modules admits small colimits. I was told to try proving that the chain complex defined degree wise as the colimit of the modules of the same degree is the colimit of a given diagram. I built the boundary operator following this hint found on MathOverflow, but now I'm stuck in proving that such complex has indeed the universal property. I took another cone $C$ (With module indexed $C_k$) over my diagram $D(n)$ (each $D(n)$ is a chain complex), and noticed that degree-wise I have unique maps $$i_k \colon \text{Colim}_n D(n)_k \to C_k$$
this is indeed my candidate for the map from the Colim to $C$ But I'm unable to prove that it is indeed a map of chain complexes!
Can someone help me with the proof?