0P5V Lemma 4.3.7. We have Eβ‘(f)βHomπ±β‘(Eβ‘(m,Ο),Eβ‘(m~,Ο~))E(f)\in\operatorname{Hom}\nolimits_{{\mathcal{V}}}(E(m,\pi),E(\tilde{m},\tilde{\pi})). The construction makes EE into a differential endofunctor of π±{\mathcal{V}}.
0P5W Proof. The lemma follows from the commutativity of the following diagram: E22β(m)βE2βE1β(m)\textstyle{E_{2}^{2}(m)\oplus E_{2}E_{1}(m)}E1βE2β(m)βE12β(m)\textstyle{E_{1}E_{2}(m)\oplus E_{1}^{2}(m)}E22β(m~)βE2βE1β(m~)\textstyle{E_{2}^{2}(\tilde{m})\oplus E_{2}E_{1}(\tilde{m})}E1βE2β(m~)βE12β(m~)\textstyle{E_{1}E_{2}(\tilde{m})\oplus E_{1}^{2}(\tilde{m})}ΟβE2βΟβΟ2\scriptstyle{\sigma\circ E_{2}\pi\circ\tau_{2}}Ο1βE1βΟβΟ\scriptstyle{\tau_{1}\circ E_{1}\pi\circ\sigma}Ο\scriptstyle{\sigma}E22βf\scriptstyle{E_{2}^{2}f}E2βE1βf\scriptstyle{E_{2}E_{1}f}E1βE2βf\scriptstyle{E_{1}E_{2}f}E12βf\scriptstyle{E_{1}^{2}f}ΟβE2βΟ~βΟ2\scriptstyle{\sigma\circ E_{2}\tilde{\pi}\circ\tau_{2}}Ο1βE1βΟ~βΟ\scriptstyle{\tau_{1}\circ E_{1}\tilde{\pi}\circ\sigma}Ο\scriptstyle{\sigma} β