Proof.
We first prove $(1)$. Let $W$ be the collection of all $\lambda _{+}$-cocartesian morphisms of $\operatorname{\mathcal{C}}$. Note that a morphism $u$ of $\operatorname{\mathcal{C}}$ belongs to $W$ if and only if $\lambda _{-}(u)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-}$ (Proposition 8.2.1.7). Suppose that, for every object $C_{-} \in \operatorname{\mathcal{C}}_{-}$, the $\infty $-category $\{ C_{-} \} \times _{\operatorname{\mathcal{C}}_{-}} \operatorname{\mathcal{C}}$ is weakly contractible. Applying Corollary 6.3.5.3, we deduce that the functor $\lambda _{-}$ exhibits $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. It follows that precomposition with $\lambda _{-}$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[ W^{-1}], \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} )$, where $\operatorname{Fun}( \operatorname{\mathcal{C}}[ W^{-1}], \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} )$ denotes the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} )$ spanned by those functors which carry each element of $W$ to an isomorphism in $\operatorname{\mathcal{D}}^{\operatorname{op}}_{-}$ (Notation 6.3.1.1). We have a commutative diagram of $\infty $-categories
8.31
\begin{equation} \begin{gathered}\label{equation:collapse-left-universal} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \ar [d]^{\circ \lambda _{-}} \\ \operatorname{\mathcal{E}}_{-} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{+} ) } \operatorname{Fun}(\operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} ) \ar [d] \\ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{+}) } \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \ar [r]^-{\theta } & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} ) } \end{gathered} \end{equation}
in which both squares are pullbacks. To prove $(1)$, it will suffice to show that the upper square is a categorical pullback diagram (Proposition 4.5.2.21). In fact, we will show that $\theta $ is an isofibration, so that both squares are categorical pullback diagrams (Corollary 4.5.2.27). This follows by observing that $\theta $ factors as a composition
\[ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}_{+} )} \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \xrightarrow {\theta '} \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} ) \times \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \xrightarrow {\theta ''} \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} ), \]
where $\theta '$ is a pullback of the composition map $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \xrightarrow { \mu \circ } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+} )$ (hence a left fibration by virtue of Corollary 4.2.5.2) and $\theta ''$ is a pullback of the projection map $\operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{D}}_{+} ) \rightarrow \Delta ^0$. This completes the proof of assertion $(1)$.
Assertion $(2)$ follows by a similar argument. We now prove $(3)$. Suppose that $\lambda $ satisfies the hypotheses of both $(1)$ and $(2)$; we wish to prove that the forgetful functor $T: \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{E}}_{\pm }$ is an equivalence of $\infty $-categories. Note that $T$ factors as a composition
\[ \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \xrightarrow {T'} \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} )} \operatorname{\mathcal{E}}_{+} \xrightarrow {T''} \operatorname{\mathcal{E}}_{\pm }, \]
where $T'$ is an equivalence of $\infty $-categories by virtue of $(2)$. It will therefore suffice to show that $T''$ is an equivalence of $\infty $-categories. We have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} )} \operatorname{\mathcal{E}}_{+} \ar [d]^{T''} \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{D}}_{-} )^{\operatorname{op}} \ar [d]^{ \circ \lambda _{-} } \\ \operatorname{\mathcal{E}}_{\pm } \ar [r]^-{\mu _{-} \circ } \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} ) \ar [d] \\ \operatorname{\mathcal{E}}_{+} \ar [r]^-{\rho } & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} ), } \]
where both squares are pullbacks and the upper right vertical map is an equivalence of $\infty $-categories. It will therefore suffice to show that the upper square is a categorical pullback diagram (Proposition 4.5.2.21). In fact, we claim that $\rho $ is an isofibration, so that both squares are categorical pullback diagrams (Corollary 4.5.2.27). This follows by observing that $\rho $ is the restriction of the map $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \xrightarrow { \mu _{-} \circ } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} )$ (which is a cocartesian fibration by virtue of Proposition 8.2.1.7 and Theorem 5.2.1.1) to a replete subcategory $\operatorname{\mathcal{E}}_{+} \subseteq \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
$\square$