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Remark 8.3.4.18 (Change of $\mathscr {K}$). Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. Suppose we are given a pair of profunctors $\mathscr {K}, \mathscr {K}': \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$, a natural transformation $\alpha : \mathscr {K} \rightarrow \mathscr {K}'$, and a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ & \underline{ \Delta ^0 }_{ \operatorname{Tw}(\operatorname{\mathcal{D}})} \ar [dl]^-{\beta } \ar [dr]_-{\beta '} & \\ \mathscr {K}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})} \ar [rr]^{ \alpha |_{ \operatorname{Tw}( \operatorname{\mathcal{D}})} } & & \mathscr {K}'|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})} } \]

in the $\infty $-category $\operatorname{Fun}( \operatorname{Tw}( \operatorname{\mathcal{D}}), \operatorname{\mathcal{S}})$. Then any two of the following conditions imply the third:

  • The natural transformation $\beta $ exhibits the profunctor $\mathscr {K}$ as represented by $G$.

  • The natural transformation $\beta '$ exhibits the profunctor $\mathscr {K}'$ as represented by $G$.

  • The natural transformation $\alpha $ is an isomorphism.

See Remark 5.6.6.4.