Consider the adjunction isomorphism
ϕ : Hom B ( E 2 L , E 1 L ) → ∼ Hom B ( E 1 ∨ E 2 L , L ) \phi:\operatorname{Hom}\nolimits_{B}(E_{2}L,E_{1}L)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits_{B}(E_{1}^{\vee}E_{2}L,L)
Let π ∈ Z Hom B ( E 2 L , E 1 L ) \pi\in Z\operatorname{Hom}\nolimits_{B}(E_{2}L,E_{1}L) ς = ϕ ( π ) ∈ Z Hom B ( E 1 ∨ E 2 L , L ) \varsigma=\phi(\pi)\in Z\operatorname{Hom}\nolimits_{B}(E_{1}^{\vee}E_{2}L,L) 5.3.2
(5.3.3) E 2 2 L \textstyle{E_{2}^{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} E 2 π \scriptstyle{E_{2}\pi} τ 2 \scriptstyle{\tau_{2}} E 2 E 1 L \textstyle{E_{2}E_{1}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} σ \scriptstyle{\sigma} E 1 E 2 L \textstyle{E_{1}E_{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} E 1 π \scriptstyle{E_{1}\pi} E 1 2 L \textstyle{E_{1}^{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} τ 1 \scriptstyle{\tau_{1}} E 2 2 L \textstyle{E_{2}^{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} E 2 π \scriptstyle{E_{2}\pi} E 2 E 1 L \textstyle{E_{2}E_{1}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} σ \scriptstyle{\sigma} E 1 E 2 L \textstyle{E_{1}E_{2}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces} E 1 π \scriptstyle{E_{1}\pi} E 1 2 L \textstyle{E_{1}^{2}L}
This gives us an identification (isomorphism of categories)
between differential A A [ L , π ] [L,\pi] L L B B π ∈ Z Hom B ( E 2 L , E 1 L ) \pi\in Z\operatorname{Hom}\nolimits_{B}(E_{2}L,E_{1}L) 5.3.3