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In older literature, such as the original work of Bousfield–Kan \cite{BousfieldKan}, the homotopy limit of a diagram $X\colon I\to sSet$ is defined as the weighted limit
$hom(I/-, X),$where the weight $I/-$ sends $i\mapsto N(I/i)$, the nerve of the comma category of $i$.
Such a functor must be derived in order to get the correct (homotopy invariant) notion, which in this case amounts to replacing $X$ with an objectwise weakly equivalent diagram valued in Kan complexes.
At some point (when?) a shift in terminology happened, and in the modern parlance homotopy limits are commonly assumed to be derived.
I have added to this addition (here) a link to Bousfield-Kan map and also to the entry’s own section “Resolved (co)ends”.
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