Variant 9.2.1.3 ($\operatorname{Pro}$-Completions). Let $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories. We say that $H$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\operatorname{Pro}$-completion of $\operatorname{\mathcal{C}}$ if it satisfies the following dual version of Definition 9.2.1.1:
- $(a')$
The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ admits small cofiltered limits.
- $(b')$
If $\operatorname{\mathcal{D}}$ is any $\infty $-category which admits small cofiltered limits, then precomposition with $H$ induces an equivalence $\operatorname{Fun}'( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, where $\operatorname{Fun}'( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ is the full subcategory of $\operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ spanned by those functors which preserve small cofiltered limits.
If these conditions are satisfied, then the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is uniquely determined up to equivalence and depends functorially on $\operatorname{\mathcal{C}}$. To emphasize this dependence, we will often denote $\widehat{\operatorname{\mathcal{C}}}$ by $\operatorname{Pro}(\operatorname{\mathcal{C}})$ and refer to it as the $\operatorname{Pro}$-completion of $\operatorname{\mathcal{C}}$. Note that we have a canonical equivalence $\operatorname{Pro}(\operatorname{\mathcal{C}})^{\operatorname{op}} \simeq \operatorname{Ind}(\operatorname{\mathcal{C}}^{\operatorname{op}})$.