Warning 7.5.1.4. In [MR0365573], Bousfield and Kan define the homotopy limit of an arbitrary diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ to be the simplicial set $\operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}))$ appearing in Construction 7.5.1.1. We will avoid this convention for two reasons:
Many important features of the Bousfield-Kan construction (such as the homotopy invariance property of Remark 7.5.1.3) are true for diagrams of Kan complexes, but not for general diagrams of simplicial sets.
In the case where $\mathscr {F}$ is a diagram of $\infty $-categories, it will be convenient to adopt a slightly different definition of homotopy limit (Construction 7.5.2.1), which generally does not agree with the Bousfield-Kan construction.