Kerodon

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Corollary 8.5.6.5. For every $\infty$-category $\operatorname{\mathcal{C}}$, evaluation on the non-identity morphism of $\operatorname{Idem}$ induces a bijection

$\theta : \pi _0( \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( (\operatorname{End}_{\operatorname{\mathcal{C}}}^{\mathrm{idm} })^{\simeq } ).$

Proof. The surjectivity of $\theta$ follows from the definition of an idempotent endomorphism, and the injectivity from Proposition 8.5.6.4. $\square$