The left-to-right exact equivalence sends the identity to the Nakayama functor

The composites ${\mathrm{\Gamma }}^{lr}{\mathrm{\Gamma }}^{rl}={\mathrm{\Phi }}^l{\mathrm{\Psi }}^r{\mathrm{\Phi }}^r{\mathrm{\Psi }}^l$ and ${\mathrm{\Gamma }}^{rl}{\mathrm{\Gamma }}^{lr}={\mathrm{\Phi }}^r{\mathrm{\Psi }}^l{\mathrm{\Phi }}^l{\mathrm{\Psi }}^r$ are computed by substituting the quasi-inverse isomorphisms of [F1]: conjugating ${\mathrm{\Psi }}^r{\mathrm{\Phi }}^r\cong 1$ by ${\mathrm{\Phi }}^l$ and ${\mathrm{\Psi }}^l$ gives ${\mathrm{\Phi }}^l{\mathrm{\Psi }}^r{\mathrm{\Phi }}^r{\mathrm{\Psi }}^l\cong {\mathrm{\Phi }}^{l}1{\mathrm{\Psi }}^l={\mathrm{\Phi }}^l{\mathrm{\Psi }}^l\cong 1$, and conjugating ${\mathrm{\Psi }}^l{\mathrm{\Phi }}^l\cong 1$ by ${\mathrm{\Phi }}^r$ and ${\mathrm{\Psi }}^r$ gives ${\mathrm{\Phi }}^r{\mathrm{\Psi }}^l{\mathrm{\Phi }}^l{\mathrm{\Psi }}^r\cong {\mathrm{\Phi }}^{r}1{\mathrm{\Psi }}^r={\mathrm{\Phi }}^r{\mathrm{\Psi }}^r\cong 1$, all by [F4]. Hence ${\mathrm{\Gamma }}^{lr}{\mathrm{\Gamma }}^{rl}\cong 1$ and ${\mathrm{\Gamma }}^{rl}{\mathrm{\Gamma }}^{lr}\cong 1$, so ${\mathrm{\Gamma }}^{rl}$ and ${\mathrm{\Gamma }}^{lr}$ are quasi-inverse to each other.

By definition [F2] one has ${\mathrm{\Gamma }}^{rl}(1_{\mathcal{A}})=N_{\mathcal{A}}^r$ and ${\mathrm{\Gamma }}^{lr}(1_{\mathcal{A}})=N_{\mathcal{A}}^l$, so ${\mathrm{\Gamma }}^{rl}$ sends the identity functor, regarded as a left exact endofunctor, to the Nakayama functor $N_{\mathcal{A}}^r$, and dually ${\mathrm{\Gamma }}^{lr}$ sends the identity, regarded as right exact, to $N_{\mathcal{A}}^l$. By [F3] these are computed in the model as $N_{\mathcal{A}}^r\cong A^{\ast }{\otimes }_A-$ and $N_{\mathcal{A}}^l\cong {\mathrm{Hom}}_A(A^{\ast },-)$.

Since $1_{\mathcal{A}}$ is exact by [F2], it is a common object of $\mathrm{Lex}(\mathcal{A},\mathcal{A})$ and $\mathrm{Rex}(\mathcal{A},\mathcal{A})$, and the natural candidate for the equivalence to agree with the identity-on-objects inclusion of the exact endofunctors is the family of isomorphisms ${\mathrm{\Gamma }}^{rl}(F)\cong F$ for the endofunctors $F$ that are both left and right exact. If such a family existed, its member at $F=1_{\mathcal{A}}$ together with ${\mathrm{\Gamma }}^{rl}(1_{\mathcal{A}})=N_{\mathcal{A}}^r$ would give a natural isomorphism $N_{\mathcal{A}}^r\cong 1_{\mathcal{A}}$ by [F4], contradicting the hypothesis that $N_{\mathcal{A}}^r$ is not naturally isomorphic to the identity. Hence this restriction of ${\mathrm{\Gamma }}^{rl}$ fails to be naturally isomorphic to the inclusion of exact endofunctors into $\mathrm{Rex}(\mathcal{A},\mathcal{A})$ whenever $N_{\mathcal{A}}^r$ is not naturally isomorphic to the identity, and in particular the equivalence between left exact and right exact endofunctors is not the identity-on-objects inclusion of the exact endofunctors in general; the companion examples page exhibits a category where the hypothesis holds. Only the finite model, the identity functor and the finitely many (co)end data defining the triangle enter, so no commutativity of $A$ and no choice are used.