Suppose I have a small category $ mathcal{C} $ which is fibered over some category $mathcal{I}$ in the categorical sense. That is, there is a functor $pi : mathcal{C} rightarrow mathcal{I}$ which is a fibration of categories. (One way to say this, I guess, is that $mathcal{C}$ has a factorization system consisting of vertical arrows, i.e. the ones that $pi$ sends to an identity arrow in $mathcal{I}$, and horizontal arrows, which are the ones it does not. But there are many other characterizations.)
Now let $F : mathcal{C} rightarrow smathcal{S}$ be a diagram of simplicial sets indexed by $mathcal{C}$. My question concerns the homotopy limit of $F$. Intuition tells me that there should be an equivalence
$$ varprojlim_{mathcal{C}} F simeq varprojlim_{mathcal{I}} left (varprojlim_{mathcal{C}_i} F_i right ) $$
where I write $mathcal{C}_i = pi^{-1}(i)$ for any $i in mathcal{I}$, $F_i$ for the restriction of $F$ to $mathcal{C}_i$ and $varprojlim$ for the homotopy limit.
Intuitively this says that when $mathcal{C}$ is fibered over $mathcal{I}$, I can find the homotopy limit of a $mathcal{C}$ diagram of spaces by first forming the homotopy limit of all the fibers, realizing that this collection has a natural $mathcal{I}$ indexing, and then taking the homotopy limit of the resulting diagram.
Does anyone know of a result like this in the model category literature?
Update: After reading the responses, I was able to find a nice set of exercises here which go through this result in its homotopy colimit version.
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