Definition 8.5.5.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that a functor of $\infty $-categories $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an idempotent completion of $\operatorname{\mathcal{C}}$ if it satisfies the following conditions:
- $(1)$
The functor $H$ is fully faithful.
- $(2)$
The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is idempotent complete.
- $(3)$
For every object $Y \in \widehat{\operatorname{\mathcal{C}}}$, there exists an object $X \in \operatorname{\mathcal{C}}$ such that $Y$ is a retract of $H(X)$.
We will say that an $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is an idempotent completion of $\operatorname{\mathcal{C}}$ if there exists a functor $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an idempotent completion of $\operatorname{\mathcal{C}}$.