# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Proof. It follows from $(a)$ that, for every $\infty$-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a map

$\theta _{\operatorname{\mathcal{E}}}: \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}[ W^{-1} ], \operatorname{\mathcal{E}})^{\simeq } )$

By virtue of Proposition 6.3.1.12, it will suffice to show that $\theta _{\operatorname{\mathcal{E}}}$ is bijective for every $\infty$-category $\operatorname{\mathcal{E}}$. Let $\rho : \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}[ W^{-1} ], \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } )$ be the map given by precomposition with $G$. It follows from $(b)$ that the composition $\rho \circ \theta _{\operatorname{\mathcal{E}}}$ is the identity. We will complete the proof by showing that $\theta _{\operatorname{\mathcal{E}}} \circ \rho$ is also the identity. Let $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ be a functor which carries each element of $W$ to an isomorphism in $\operatorname{\mathcal{E}}$. By virtue of Theorem 4.4.4.4, $H$ carries the natural transformation $u$ to an isomorphism $H = H \circ \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow H \circ G \circ F$ in the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$, so that the isomorphism class of $H$ is fixed by the composition $\theta _{\operatorname{\mathcal{E}}} \circ \rho$. $\square$