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. 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.