Day畳み込みの保存量

プロ函手の自然変換

$(F\star G)_\mathscr{D}^\mathscr{C},R_\mathscr{D}^\mathscr{C}\in\mathbb{Prof}(\mathscr{C,D})$の間の自然変換を考える．
$$ \begin{align} \text{Nat}((F\star G)_\mathscr{D}^\mathscr{C},R_\mathscr{D}^\mathscr{C})&=\overline{\bigotimes_{\substack{C\in\mathscr{C}\\D\in\mathscr{D}}}}\text{Hom}\left((F\star G)_{(D)}^{(C)},R_{(D)}^{(C)}\right)\\ &=\overline{\bigotimes_{\substack{C\in\mathscr{C}\\D\in\mathscr{D}}}}\text{Hom}\left(\overline{\bigoplus_{\substack{W,Y\in\mathscr{C}\\X,Z\in\mathscr{D}}}}F^{(W)}_{(X)}G^{(Y)}_{(Z)}\Delta_{(W\otimes Y)}^{(C)}\Delta^{(X\otimes Z)}_{(D)},R_{(D)}^{(C)}\right)\\ &=\overline{\bigotimes_{\substack{C,W,Y\in\mathscr{C}\\D,X,Z\in\mathscr{D}}}}\text{Hom}\left(F^{(W)}_{(X)}G^{(Y)}_{(Z)}\Delta_{(W\otimes Y)}^{(C)}\Delta^{(X\otimes Z)}_{(D)},R_{(D)}^{(C)}\right)\\ &\simeq\overline{\bigotimes_{\substack{C,W,Y\in\mathscr{C}\\D,X,Z\in\mathscr{D}}}}\text{Hom}\left(F^{(W)}_{(X)}G^{(Y)}_{(Z)},R_{(X\otimes Z)}^{(W\otimes Y)}\right)\\ &=\text{Nat}\left(F_\mathscr{D}^\mathscr{C}G_\mathscr{D}^\mathscr{C},R_{(\mathscr{D}\otimes \mathscr{D})}^{(\mathscr{C}\otimes \mathscr{C})}\right)\\ \end{align} $$
つまりDay畳み込み$(*)$は自身の自然変換をプロ函手の積との間で保存する．