Lemma 8.2.3.6. Suppose we are given a morphism of couplings
8.36
\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:
- $(1)$
The diagram (8.36) is a categorical pullback square.
- $(2)$
For every object $Y \in \operatorname{\mathcal{C}}_{+}$, the functor $F$ induces an equivalence of $\infty $-categories
\[ F_{Y}: \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}_{+} } \{ Y \} \rightarrow \operatorname{\mathcal{D}}\times _{ \operatorname{\mathcal{D}}_{+} } \{ F_{+}(Y) \} . \]- $(3)$
For every universal object $C \in \operatorname{\mathcal{C}}$, the image $F(C) \in \operatorname{\mathcal{D}}$ is universal.