Remark 7.5.3.15 (The Homotopy Limit as a Right Derived Functor). The results of this section can be interpreted in the language of model categories. For every small category $\operatorname{\mathcal{C}}$, the category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ can be equipped with a model structure in which the cofibrations are monomorphisms and the weak equivalences are levelwise categorical equivalences (see Example 
). The inverse limit functor
then admit a right derived functor $\underset {\longleftarrow }{\mathrm{Rlim}}: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow \operatorname{Set_{\Delta }}$, which carries a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ to the limit of a fibrant replacement of $\mathscr {F}$. It follows from Propositions 7.5.3.4 and 7.5.3.7 that, when restricted to the subcategory $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{QCat}) \subset \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$, the functor $\underset {\longleftarrow }{\mathrm{Rlim}}$ is (categorically) equivalent to the homotopy limit functor $ \underset {\longleftarrow }{\mathrm{holim}}: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{QCat}) \rightarrow \operatorname{QCat}$ of Construction 7.5.2.1. We will return to this point in ยง.