I am looking for a reference, preferably as elementary as possible, for the following statement. 


Let $X\_{m,n}$ be a bi-simplicial object in an **additive** category $\mathcal{A}$. Then the complex $|X\_{m,n}|$ obtained from iterative realization (i.e. the total complex of the associated bi-complex) is homotopy equivalent to the complex obtained by realizing the diagonal $|X\_{n,n}|$. 


I know that it should follow from the cofinality of the map $\Delta^{op} \to \Delta^{op} \times \Delta^{op}$ together with the identification of the realization as the colimit of a simplicial object in the $\infty$-category of bounded above complexes in $\mathcal{A}$. However, all this seem to me like a real overkill for this problem, and I would expect a much more elementary explanation can be given. I also remember seeing a proof along this lines in the case where $\mathcal{A}$ is Abelian, but I really need a more general result, since the example I have in mind is not abelian. 



Simplicial Objects in Additive Categories
For abelian categories this is known as the Eilenberg–Zilber theorem, see, for instance, Theorem 8.5.1 in Weibel's book. One can write down explicit comparison maps in both directions (namely, the Alexander–Whitney and Eilenberg–Zilber maps) and also write down an explicit chain homotopy that shows these maps to be chain homotopy equivalences. The constructions are explicit and concrete, and perhaps the proof goes through for additive categories, but I have not checked this.


