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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?enter image description here

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    $\begingroup$ The question is just about how to define $\Delta$ without elements ? This is easy then, $\Delta$ is the difference between the identity $\prod_{i\in\mathbb{N}}A_i\to\prod_{i\in\mathbb{N}}A_i$ and the map such that on the $A_j$ component, it is $\prod_{i\in\mathbb{N}}A_i\to A_{j+1}\to A_j$. $\endgroup$ – Roland Jan 6 at 12:32
  • $\begingroup$ Thanks, it was so silly of me to miss that! $\endgroup$ – Jehu314 Jan 6 at 14:04

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