Definition 5.3.5.1. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}_0, \mathscr {F}_{1}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be diagrams of $\infty $-categories indexed by $\operatorname{\mathcal{C}}$. We will say that $\mathscr {F}_0$ and $\mathscr {F}_1$ are levelwise equivalent if there exists another diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ equipped with levelwise categorical equivalences $\mathscr {F}_0 \rightarrow \mathscr {F} \leftarrow \mathscr {F}_1$ (see Definition 4.5.6.1).
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