Remark 5.6.5.19 (Rectification). Corollary 5.6.5.18 is a prototypical example of a rectification result. If $\operatorname{\mathcal{C}}$ is an ordinary category, then a functor of $\infty $-categories $\mathscr {F}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$ can be viewed as a homotopy coherent diagram in the simplicial category $\operatorname{QCat}$:
To every object $X$ of the category $\operatorname{\mathcal{C}}$, the functor $\mathscr {F}$ associates an $\infty $-category $\mathscr {F}(X)$.
To every morphism $u: X \rightarrow Y$ of the category $\operatorname{\mathcal{C}}$, the functor $\mathscr {F}$ associates a functor of $\infty $-categories $\mathscr {F}(u): \mathscr {F}(X) \rightarrow \mathscr {F}(Y)$.
To every pair of composable morphisms $u: X \rightarrow Y$ and $v: Y \rightarrow Z$ in the category $\operatorname{\mathcal{C}}$, the functor $\mathscr {F}$ associates an isomorphism of functors $\alpha _{u,v}: \mathscr {F}(v) \circ \mathscr {F}(u) \rightarrow \mathscr {F}(v \circ u)$.
When applied to higher-dimensional simplices of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the functor $\mathscr {F}$ provides additional data which encode coherence laws satisfied by the isomorphisms $\alpha _{u,v}$.
Corollary 5.6.5.18 asserts that we can always find a strictly commutative diagram $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ which is isomorphic to $\mathscr {F}$ in the $\infty $-category $\operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{QC}})$. In particular, the diagram $\mathscr {G}$ carries each object $X \in \operatorname{\mathcal{C}}$ to an $\infty $-category $\mathscr {G}(X)$ which is equivalent to $\mathscr {F}(X)$ (beware that we generally cannot arrange that $\mathscr {G}(X)$ is isomorphic to $\mathscr {F}(X)$ as a simplicial set).
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, we will prove a more refined version of this result, which allows us to describe the entire $\infty $-category $\operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{QC}})$ in terms of strictly commutative diagrams indexed by $\operatorname{\mathcal{C}}$ (Proposition ).