Proof.
Fix an uncountable regular cardinal $\kappa $ for which both $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are essentially $\kappa $-small. Precomposition with $F$ determines a functor $F^{\ast }: \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. It follows from Proposition 7.6.6.13 that the functor $F^{\ast }$ admits a left adjoint $F_{!}$ (given by left Kan extension along $F^{\operatorname{op}}$). Let $h^{\operatorname{\mathcal{C}}}_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ and $h^{\operatorname{\mathcal{D}}}_{\bullet }: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ be covariant Yoneda embeddings for $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively, so that the diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{F} \ar [r]^-{h^{\operatorname{\mathcal{C}}}_{\bullet } } & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \ar [d]^{ F_{!} } \\ \operatorname{\mathcal{D}}\ar [r]^-{ h^{\operatorname{\mathcal{D}}}_{\bullet } } & \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) } \]
commutes up to isomorphism (Example 8.4.4.5), where the horizontal maps are fully faithful (Theorem 8.3.3.13). It follows that, for every pair of objects $C', C \in \operatorname{\mathcal{C}}$, we can identify $\theta _{C',C}$ with the comparison map
\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) }( h^{\operatorname{\mathcal{C}}}_{C'}, h^{\operatorname{\mathcal{C}}}_{C} ) & \rightarrow & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) }( F_{!} h^{\operatorname{\mathcal{C}}}_{C'}, F_{!} h^{\operatorname{\mathcal{C}}}_{C} ) \\ & \simeq & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) }( h^{\operatorname{\mathcal{C}}}_{C'}, F^{\ast } F_{!} h^{\operatorname{\mathcal{C}}}_{C} ) \end{eqnarray*}
given by precomposition with the unit $u: h^{\operatorname{\mathcal{C}}}_{C} \rightarrow F^{\ast } F_{!} h^{\operatorname{\mathcal{C}}}_{C}$. Combining this observation with Proposition 8.3.1.1, we see that condition $(2)$ can be restated as follows:
- $(2')$
The unit map $u: h^{\operatorname{\mathcal{C}}}_{C} \rightarrow F^{\ast } F_{!} h^{\operatorname{\mathcal{C}}}_{C}$ is an isomorphism in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.
Our assumption that $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ guarantees that the pullback functor $F^{\ast }$ is fully faithful. Using Remark 6.2.2.20, we see that $u$ is an isomorphism if and only the representable functor $h_{C}^{\operatorname{\mathcal{C}}}$ belongs to the essential image of $F^{\ast }$: that is, the collection of functors $\operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ which carry every morphism of $W$ to an isomorphism in $\operatorname{\mathcal{S}}^{< \kappa }$. This is a reformulation of $(1)$.
$\square$