A Remark in Weibel's "Introduction to Homological Algebra"

Question

In the section on the derived functors of the inverse limit(with $...3\rightarrow 2 \rightarrow 1 \rightarrow 0$ as index category), Weibel constructs the inverse limit using the map $\Delta$ in the photograph. Later, he remarks that everything goes through for any $AB4*$ category. But I can't even construct the map in the general case. My question is, how does one do this for any $AB4*$ category?