Remark 9.4.5.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then $F$ is a Morita equivalence if and only if it satisfies the following condition:
- $(\ast )$
For every idempotent complete $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a bijection of sets
\[ \pi _0( \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})^{\simeq } ). \]
The necessity of condition $(\ast )$ is immediate. Conversely, suppose that condition $(\ast )$ is satisfied, and let $\operatorname{\mathcal{E}}$ be an idempotent complete $\infty $-category; we wish to show that the functor $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \xrightarrow { \circ F} \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is an equivalence of $\infty $-categories. By virtue of Proposition 4.5.1.22, it will suffice to show that for every simplicial set $K$, the induced map
is a bijection. This follows by applying condition $(\ast )$ to the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{E}})$ (which is idempotent complete by virtue of Corollary 8.5.4.10).