# Kerodon

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Example 8.2.2.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. Then the commutative diagram

8.33
$$\begin{gathered}\label{equation:example:easy-corepresentable-hom} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{ \operatorname{Tw}(F) } \ar [d] & \operatorname{Tw}(\operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\ar [r]^-{ F^{\operatorname{op}} \times F } & \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}} \end{gathered}$$

is a morphism of couplings. It follows from Theorem 8.2.2.11 that this morphism is initial when viewed as an object of the $\infty$-category $\{ F \} \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} } \operatorname{Fun}_{\pm }( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{Tw}(\operatorname{\mathcal{D}}) )$.