Left and right Nakayama functors by finite kernel calculus

Let $\mathcal{A}$ be a finite $k$-linear abelian category (Finite k-linear abelian categories, Abelian category, k-linear categories and k-linear functors) and let ${\mathrm{\Phi }}^l,{\mathrm{\Phi }}^r,{\mathrm{\Psi }}^l,{\mathrm{\Psi }}^r$ be the equivalences of the categorical Eilenberg–Watts triangle (Categorical Eilenberg–Watts equivalences for finite linear categories, Finite Eilenberg–Watts kernels: explicit end and coend universal maps). Define ${\mathrm{\Gamma }}^{rl}={\mathrm{\Phi }}^r{\mathrm{\Psi }}^l:\mathrm{Lex}(\mathcal{A},\mathcal{A})⟶\mathrm{Rex}(\mathcal{A},\mathcal{A}),{\mathrm{\Gamma }}^{lr}={\mathrm{\Phi }}^l{\mathrm{\Psi }}^r:\mathrm{Rex}(\mathcal{A},\mathcal{A})⟶\mathrm{Lex}(\mathcal{A},\mathcal{A})$ (using the notation of Left exact and right exact functors and Natural transformation and its components). The Nakayama functor of $\mathcal{A}$ is $N_{\mathcal{A}}^r={\mathrm{\Gamma }}^{rl}(1_{\mathcal{A}})$, the image of the identity functor regarded as a left exact endofunctor, and its left exact analogue is $N_{\mathcal{A}}^l={\mathrm{\Gamma }}^{lr}(1_{\mathcal{A}})$, the identity regarded as a right exact endofunctor (Covariant functor, identity functor, composite functor, and contravariant functor). In a module model $\mathcal{A}\simeq A\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ these are the endofunctors $N^r\cong A^{\ast }{\otimes }_A-$ and $N^l\cong {\mathrm{Hom}}_A(A^{\ast },-)$. The definition asserts no further properties, selects no object and makes no choice; well-definedness, independence of the module model, the intrinsic (co)end formulas and the adjunction $N^r⊣N^l$ are proved in Nakayama kernels give well-defined adjoint functors ↗.

Let $\mathcal{A}$ be a finite $k$-linear abelian category (Finite k-linear abelian categories, Abelian category, k-linear categories and k-linear functors) and let ${\mathrm{\Phi }}^l,{\mathrm{\Phi }}^r,{\mathrm{\Psi }}^l,{\mathrm{\Psi }}^r$ be the equivalences of the categorical Eilenberg–Watts triangle (Categorical Eilenberg–Watts equivalences for finite linear categories, Finite Eilenberg–Watts kernels: explicit end and coend universal maps). Define ${\mathrm{\Gamma }}^{rl}={\mathrm{\Phi }}^r{\mathrm{\Psi }}^l:\mathrm{Lex}(\mathcal{A},\mathcal{A})⟶\mathrm{Rex}(\mathcal{A},\mathcal{A}),{\mathrm{\Gamma }}^{lr}={\mathrm{\Phi }}^l{\mathrm{\Psi }}^r:\mathrm{Rex}(\mathcal{A},\mathcal{A})⟶\mathrm{Lex}(\mathcal{A},\mathcal{A})$ (using the notation of Left exact and right exact functors and Natural transformation and its components). The Nakayama functor of $\mathcal{A}$ is $N_{\mathcal{A}}^r={\mathrm{\Gamma }}^{rl}(1_{\mathcal{A}})$, the image of the identity functor regarded as a left exact endofunctor, and its left exact analogue is $N_{\mathcal{A}}^l={\mathrm{\Gamma }}^{lr}(1_{\mathcal{A}})$, the identity regarded as a right exact endofunctor (Covariant functor, identity functor, composite functor, and contravariant functor). In a module model $\mathcal{A}\simeq A\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ these are the endofunctors $N^r\cong A^{\ast }{\otimes }_A-$ and $N^l\cong {\mathrm{Hom}}_A(A^{\ast },-)$. The definition asserts no further properties, selects no object and makes no choice; well-definedness, independence of the module model, the intrinsic (co)end formulas and the adjunction $N^r⊣N^l$ are proved in Nakayama kernels give well-defined adjoint functors ↗.

• Why the composites are legitimate. ${\mathrm{\Psi }}^l$ is defined on $\mathrm{Lex}(\mathcal{A},\mathcal{A})$ with values in ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}⊠\mathcal{A}$ and ${\mathrm{\Phi }}^r$ is defined on ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}⊠\mathcal{A}$ with values in $\mathrm{Rex}(\mathcal{A},\mathcal{A})$, so the composite ${\mathrm{\Gamma }}^{rl}$ is a functor on the functor category of left exact endofunctors with all natural transformations; dually ${\mathrm{\Gamma }}^{lr}$ is defined on $\mathrm{Rex}(\mathcal{A},\mathcal{A})$. The identity functor is both left exact and right exact, so both evaluations $N_{\mathcal{A}}^r={\mathrm{\Gamma }}^{rl}(1_{\mathcal{A}})$ and $N_{\mathcal{A}}^l={\mathrm{\Gamma }}^{lr}(1_{\mathcal{A}})$ are legitimate and use the same object $1_{\mathcal{A}}$ in the two different functor categories.

• What the definition does not assert. No formula, adjunction, self-injectivity, symmetry or coincidence of $N^r$ and $N^l$ is asserted here: the displayed module-model formulas $N^r\cong A^{\ast }{\otimes }_A-$ and $N^l\cong {\mathrm{Hom}}_A(A^{\ast },-)$ are theorems of the justified_by supplier Nakayama kernels give well-defined adjoint functors ↗, together with the behaviour under a change of module model. In particular the definition does not choose a module model, a presentation or a basis, and it does not identify ${\mathrm{\Gamma }}^{rl}$ or ${\mathrm{\Gamma }}^{lr}$ with the identity.