Proof.
Note that the functor $U_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}$ is automatically fully faithful (Remark 7.1.5.3). Our assumption that $U_0$ is essentially surjective then guarantees that it is an equivalence of $\infty $-categories, which proves $(1)$.
We next prove the following:
- $(\ast )$
For every object $Y \in \operatorname{\mathcal{C}}$, there exists a morphism $e: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, where $X$ is $U$-initial and $U(e)$ is an isomorphism in $\operatorname{\mathcal{D}}$.
To prove $(\ast )$, we observe that the essential surjectivity of $U_0$ guarantees that there exists a $U$-initial object $X \in \operatorname{\mathcal{C}}$ and an isomorphism $\overline{e}: U(X) \rightarrow U(Y)$ in the $\infty $-category $\operatorname{\mathcal{D}}$. Since $X$ is $U$-initial, the functor $U$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(X), U(Y) )$. Modifying $\overline{e}$ by a homotopy, we can assume without loss of generality that $\overline{e} = U(e)$ for some morphism $X \rightarrow Y$ of $\operatorname{\mathcal{C}}$.
We now prove $(2)$. Let $e: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, where the object $X$ is $U$-initial. Assume first that $U(e)$ is an isomorphism in $\operatorname{\mathcal{D}}$. We wish to show that, for every $U$-initial object $C \in \operatorname{\mathcal{C}}$, postcomposition with $e$ induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C,X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y)$. This follows by inspecting the commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, X) \ar [r]^-{\circ [e] } \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(C), U(X) ) \ar [r]^-{ \circ [U(e)] } & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(C), U(Y) ) } \]
in the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$; here the vertical maps are homotopy equivalences by virtue of our assumption that $C$ is $U$-initial, and the bottom horizontal map is a homotopy equivalence by virtue of our assumption that $U(e)$ is an isomorphism.
We now prove the converse. Assume that $e: X \rightarrow Y$ exhibits $X$ as a $\operatorname{\mathcal{C}}_0$-coreflection of $Y$; we wish to show that $U(e)$ is an isomorphism. Using $(\ast )$, we can choose a $U$-initial object $X' \in \operatorname{\mathcal{C}}$ and a morphism $e': X' \rightarrow Y$ such that $U(e')$ is an isomorphism in $\operatorname{\mathcal{D}}$. It follows from the previous step that $e'$ exhibits $X'$ as a $\operatorname{\mathcal{C}}_0$-coreflection of $Y$. It follows that $e$ can be realized as the composition of $e'$ with an isomorphism $v: X \rightarrow X'$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (Remark 6.2.2.3). Then $U(e)$ is a composition of the isomorphisms $U(v)$ and $U(e')$ in the $\infty $-category $\operatorname{\mathcal{D}}$, and is therefore also an isomorphism.
Assertion $(3)$ follows immediately from $(2)$ and $(\ast )$. Combining $(3)$ with Proposition 6.2.2.13, we see that there exists a functor $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$ and a natural transformation $\eta : L \rightarrow \operatorname{id}_{\operatorname{\mathcal{C}}}$ which exhibits $L$ as a $\operatorname{\mathcal{C}}_0$-coreflection functor: that is, it carries each object $Y \in \operatorname{\mathcal{C}}$ to a morphism $\eta _{Y}: L(Y) \rightarrow Y$ where $L(Y)$ is $U$-initial and $U( \eta _ Y)$ is an isomorphism. In particular, $\eta $ induces an isomorphism $U_0 \circ L \rightarrow U$ in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ (Theorem 4.4.4.4). It follows from assumption $(1)$ that the functor $U_0$ admits a homotopy inverse $F_0: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}_0$, which is also a left adjoint of $U_0$ (Example 6.2.1.11). Moreover, the inclusion functor $\iota : \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ is left adjoint to $L$ (Proposition 6.2.2.17). It follows that the composition $F = \iota \circ F_0$ is left adjoint to $U_0 \circ L$ (Remark 6.2.1.8), and therefore also to $U$. This proves $(4)$. Moreover, the functor $F$ is fully faithful (since $F_0$ is an equivalence of $\infty $-categories and $\iota $ is the inclusion of a full subcategory), so assertion $(5)$ follows from Proposition 6.3.3.6.
$\square$