Take the 2-minute tour ×  How can I figure out the classical
construction (direct sum, product, pullbacks, and in general
direct and inverse limits) in the category made by chain
complexes and chain maps (of abelian groups or any abelian
stuff)? Because of this category is abelian it must have
(co)limits, isn't it?  In particular take $$ \begin{gather*}
\dots\to A_n\to A_{n-1}\to\dots\\ \dots\to B_n\to
B_{n-1}\to\dots \end{gather*} $$ I would like to say that
$\mathcal A\oplus\mathcal B$ is "what I want it to be",
$A_n\oplus B_n$ with obvious maps. But a standard argument
doesn't allow me to conclude it: maybe it is false? A single
word with the right reference will be enough to close the topic;
I am now reading Hilton & Stammbach.  Edit: I would like to add
what I've tried to do, but seems difficult not to invoke some
diagrams, which I'm not able to draw without a suitable package
here. However, first of all both complexes inject in the sum by
maps which are part of a chian complex, $\iota_n^A,\iota_n^B$.
Th