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

Lemma Suppose we are given a morphism of couplings

\begin{equation} \begin{gathered}\label{equation:duality-functor-left-to-right-preliminary} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{ F } \ar [d]^{\lambda } & \operatorname{\mathcal{D}}\ar [d]^{\mu } \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ F_{-}^{\operatorname{op}} \times F_{+} } & \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+}, } \end{gathered} \end{equation}

where $\lambda $ is representable and $F_{-}$ is an equivalence of $\infty $-categories. The following conditions are equivalent:


The diagram (8.36) is a categorical pullback square.


For every object $Y \in \operatorname{\mathcal{C}}_{+}$, the functor $F$ induces an equivalence of $\infty $-categories

\[ F_{Y}: \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}_{+} } \{ Y \} \rightarrow \operatorname{\mathcal{D}}\times _{ \operatorname{\mathcal{D}}_{+} } \{ F_{+}(Y) \} . \]

For every universal object $C \in \operatorname{\mathcal{C}}$, the image $F(C) \in \operatorname{\mathcal{D}}$ is universal.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Theorem and Remark To complete the proof, it will suffice to show that for each object $Y \in \operatorname{\mathcal{C}}_{+}$, the following conditions are equivalent:

$(2_ Y)$

The functor $F_{Y}$ is an equivalence of $\infty $-categories.

$(3_ Y)$

The functor $F_{Y}$ carries initial objects of $\operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}_{+} } \{ Y \} $ to initial objects of $\operatorname{\mathcal{D}}\times _{ \operatorname{\mathcal{D}}_{+} } \{ F_{+}(Y) \} $.

Replacing $\operatorname{\mathcal{D}}$ by the $\infty $-category $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times _{\operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}$, we can assume that $F_{-}$ is an isomorphism. In this case, the equivalence of $(2_ Y)$ and $(3_ Y)$ is a special case of Corollary $\square$