Proof.
Let $X$ be an object of $\operatorname{\mathcal{E}}$ having image $C = U(X)$ in $\operatorname{\mathcal{C}}$. It follows from $(a)$ that there exists a $\operatorname{\mathcal{E}}'_{C}$-local equivalence $w: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$, where $Y$ belongs to $\operatorname{\mathcal{E}}'_{C}$. To complete the proof, it will suffice to show that $w$ is a $\operatorname{\mathcal{E}}'$-local equivalence (and therefore exhibits $Y$ as a $\operatorname{\mathcal{E}}'$-localization of $X$). Fix an object $Z \in \operatorname{\mathcal{E}}'$ having image $D = U(Z)$ in $\operatorname{\mathcal{C}}$; we wish to show that precomposition with $w$ induces a homotopy equivalence $\theta : \operatorname{Hom}_{\operatorname{\mathcal{E}}}( Y, Z ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z)$. As in the proof of Proposition 6.2.4.1, we can identify $\theta $ with the upper horizontal map of a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \{ w\} \times _{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y) } \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y,Z) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z) \ar [d] \\ \{ \operatorname{id}_{C} \} \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C,C) } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C,C,D ) \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C,D ), } \]
where the lower horizontal map is a homotopy equivalence and the vertical maps are Kan fibrations (see Propositions 4.6.1.21 and 4.6.9.4). To show that this map is a homotopy equivalence, it will suffice to show that it restricts to a homotopy equivalence of vertical fibers over any vertex $\sigma \in \{ \operatorname{id}_{C} \} \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C,C) } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C,C,D )$ (Proposition 3.2.8.1). Since $\overline{\theta }$ is a homotopy equivalence, we may assume without loss of generality that $\sigma $ corresponds to a left-degenerate $2$-simplex of $\operatorname{\mathcal{C}}$, given by $s^{1}_{0}(f)$ for some morphism $f: C \rightarrow D$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Replacing $\operatorname{\mathcal{E}}$ by the fiber product $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, we are reduced to proving that $\theta $ is a homotopy equivalence in the special case where $\operatorname{\mathcal{C}}= \Delta ^1$, $C = 0$, and $D = 1$.
Let $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ be a covariant transport functor for the cocartesian fibration $U$. Then we have a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ X \ar [d]^{u} \ar [r] & f_{!}(X) \ar [d]^{f_{!}(w)} \\ Y \ar [r] & f_{!}(Y), } \]
where the horizontal maps are $U$-cocartesian. We therefore obtain a homotopy commutative diagram of morphism spaces
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z) & \operatorname{Hom}_{\operatorname{\mathcal{E}}_{D}}( f_{!}(X), Z) \ar [l] \\ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(Y,Z) \ar [u]_{ \circ [w] } & \operatorname{Hom}_{\operatorname{\mathcal{E}}_{D}}( f_{!}(Y), Z), \ar [l] \ar [u]^{ \circ [f_{!}(w)]} } \]
where the horizontal maps are homotopy equivalences (Corollary 5.1.2.3). Consequently, to show that $\theta $ is a homotopy equivalence, it will suffice to show that $f_{!}(w)$ is a $\operatorname{\mathcal{E}}'_{D}$-local equivalence in the $\infty $-category $\operatorname{\mathcal{E}}_{D}$, which follows from assumption $(b)$.
$\square$