Proposition 9.4.5.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is a Morita equivalence if and only if it satisfies the following pair of conditions:
The functor $F$ is fully faithful.
For every object $Y \in \operatorname{\mathcal{D}}$, there exists an object $X \in \operatorname{\mathcal{C}}$ such that $Y$ is a retract of $F(X)$.
Proof. Using Corollary 8.5.5.3, we can choose a functor $H: \operatorname{\mathcal{D}}\rightarrow \widehat{\operatorname{\mathcal{D}}}$ which exhibits $\widehat{\operatorname{\mathcal{D}}}$ as an idempotent completion of $\operatorname{\mathcal{D}}$. Then $H$ is a Morita equivalence (Example 9.4.5.3). By virtue of Remark 9.4.5.5, we can replace $F$ by the composite functor $H \circ F$ and thereby reduce to proving Proposition 9.4.5.8 in the special case where $\operatorname{\mathcal{D}}$ is idempotent complete. In this case, the desired result is a reformulation of Proposition 8.5.5.2. $\square$