Warning 6.3.2.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{E}}$ be an $\infty $-category. We have now given two different definitions for the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$:
- $(1)$
According to Notation 6.3.1.1, $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$ denotes the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ spanned by those diagrams $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ which carry each edge of $W$ to an isomorphism in $\operatorname{\mathcal{E}}$.
- $(2)$
By the convention of Remark 6.3.2.2, $\operatorname{\mathcal{C}}[W^{-1}]$ denotes an $\infty $-category equipped with a diagram $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ which exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. We can then consider the $\infty $-category of functors from $\operatorname{\mathcal{C}}[W^{-1}]$ to $\operatorname{\mathcal{E}}$, which we will temporarily denote by $\operatorname{Fun}'( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$.
Beware that these $\infty $-categories are not identical. However, they are equivalent: if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$, then composition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}'( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$ (Proposition 6.3.1.13). Note that the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$ does not depend on any auxiliary choices: it is well-defined up to equality as a simplicial subset of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. By contrast, the $\infty $-category $\operatorname{Fun}'( \operatorname{\mathcal{C}}[W^{-1}],\operatorname{\mathcal{E}})$ depends on the choice of the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ (and is therefore well-defined up to equivalence, but not up to isomorphism).