Day畳み込みの保存量

プロ函手の自然変換

$(F\star G{)}_{\mathcal{D}}^{\mathcal{C}},R_{\mathcal{D}}^{\mathcal{C}}\in \mathbb{Prof}(\mathcal{C}\mathcal{,}\mathcal{D})$の間の自然変換を考える．
$$\begin{array}{rl}\text{Nat}((F\star G{)}_{\mathcal{D}}^{\mathcal{C}},R_{\mathcal{D}}^{\mathcal{C}})& =\overline{\underset{\begin{array}{c}C\in \mathcal{C}\\ D\in \mathcal{D}\end{array}}{⨂}}\text{Hom}((F\star G{)}_{(D)}^{(C)},R_{(D)}^{(C)})\\ & =\overline{\underset{\begin{array}{c}C\in \mathcal{C}\\ D\in \mathcal{D}\end{array}}{⨂}}\text{Hom}(\overline{\underset{\begin{array}{c}W,Y\in \mathcal{C}\\ X,Z\in \mathcal{D}\end{array}}{⨁}}{F}_{(X)}^{(W)}{G}_{(Z)}^{(Y)}{\mathrm{\Delta }}_{(W\otimes Y)}^{(C)}{\mathrm{\Delta }}_{(D)}^{(X\otimes Z)},R_{(D)}^{(C)})\\ & =\overline{\underset{\begin{array}{c}C,W,Y\in \mathcal{C}\\ D,X,Z\in \mathcal{D}\end{array}}{⨂}}\text{Hom}(F_{(X)}^{(W)}{G}_{(Z)}^{(Y)}{\mathrm{\Delta }}_{(W\otimes Y)}^{(C)}{\mathrm{\Delta }}_{(D)}^{(X\otimes Z)},R_{(D)}^{(C)})\\ & \simeq \overline{\underset{\begin{array}{c}C,W,Y\in \mathcal{C}\\ D,X,Z\in \mathcal{D}\end{array}}{⨂}}\text{Hom}(F_{(X)}^{(W)}{G}_{(Z)}^{(Y)},R_{(X\otimes Z)}^{(W\otimes Y)})\\ & =\text{Nat}(F_{\mathcal{D}}^{\mathcal{C}}{G}_{\mathcal{D}}^{\mathcal{C}},R_{(\mathcal{D}\otimes \mathcal{D})}^{(\mathcal{C}\otimes \mathcal{C})})\end{array}$$
つまりDay畳み込み$(\ast )$は自身の自然変換をプロ函手の積との間で保存する．