Nakayama kernels give well-defined adjoint functors

The identity functor of $\mathcal{A}$ preserves every limit and colimit that exists, so it is both left exact and right exact and defines an object of $\mathrm{Lex}(\mathcal{A},\mathcal{A})$ and of $\mathrm{Rex}(\mathcal{A},\mathcal{A})$ (Left exact and right exact functors, Covariant functor, identity functor, composite functor, and contravariant functor). The composites ${\mathrm{\Gamma }}^{rl}={\mathrm{\Phi }}^r{\mathrm{\Psi }}^l$ and ${\mathrm{\Gamma }}^{lr}={\mathrm{\Phi }}^l{\mathrm{\Psi }}^r$ are composites of the equivalences of [F1], hence are themselves equivalences of categories; therefore $N_{\mathcal{A}}^r={\mathrm{\Gamma }}^{rl}(1_{\mathcal{A}})$ and $N_{\mathcal{A}}^l={\mathrm{\Gamma }}^{lr}(1_{\mathcal{A}})$ are well-defined endofunctors of $\mathcal{A}$, determined by $\mathcal{A}$, the two equivalences and the identity functor alone.

In the chosen model one has $1_{\mathcal{A}}\cong {\mathrm{\Phi }}^l(A^{\ast })$, because ${\mathrm{\Phi }}^l(A^{\ast })(X)={\mathrm{Hom}}_A((A^{\ast }{)}^{\ast },X)\cong {\mathrm{Hom}}_A(A,X)\cong X$ by [F1], the double duality of [F5] and the identification ${\mathrm{Hom}}_A(A,X)\cong X$, $f\mapsto f(1_A)$, with inverse $x\mapsto (a\mapsto ax)$; similarly $1_{\mathcal{A}}\cong {\mathrm{\Phi }}^r(A)$ because ${\mathrm{\Phi }}^r(A)(X)=A{\otimes }_{A}X\cong X$ by [F1] and [F6]. Applying the quasi-inverse isomorphisms of [F3] gives ${\mathrm{\Psi }}^l(1_{\mathcal{A}})\cong {\mathrm{\Psi }}^l{\mathrm{\Phi }}^l(A^{\ast })\cong A^{\ast }$ and ${\mathrm{\Psi }}^r(1_{\mathcal{A}})\cong {\mathrm{\Psi }}^r{\mathrm{\Phi }}^r(A)\cong A$; hence in the model $N_{\mathcal{A}}^r\cong {\mathrm{\Phi }}^r(A^{\ast })=A^{\ast }{\otimes }_A-$ and $N_{\mathcal{A}}^l\cong {\mathrm{\Phi }}^l(A)={\mathrm{Hom}}_A(A^{\ast },-)$.

Evaluation at a fixed finite module $X$ preserves the needed universal objects. In the kernel model, evaluation on Rex is $E_X^r(M)=M{\otimes }_{A}X$. For a left $A$-module $Y$, the space ${\mathrm{Hom}}_k(X,Y)$ is an $(A,A)$-bimodule with $(af)(x)=af(x)$ and $(f\cdot a)(x)=f(ax)$. Currying and its inverse $h\mapsto (m\otimes x\mapsto h(m)(x))$ give ${\mathrm{Hom}}_A(M{\otimes }_{A}X,Y)\cong {\mathrm{Hom}}_{A\text{-}A}(M,{\mathrm{Hom}}_k(X,Y))\mathrm{.}$ Thus $E_X^r$ is a left adjoint. Evaluation on Lex is $E_X^l(M)={\mathrm{Hom}}_A(M^{\ast },X)\cong {\mathrm{Hom}}_{A^{\mathrm{o}\mathrm{p}}}(X^{\ast },M)$, the isomorphism sending $f$ to $f^{\ast }$ followed by $M^{\ast \ast }\cong M$ [F5]. The mutually inverse assignments $g\mapsto (y\mapsto (\lambda \mapsto g(y\otimes \lambda )))$ and $h\mapsto (y\otimes \lambda \mapsto h(y)(\lambda ))$ give ${\mathrm{Hom}}_{A\text{-}A}(Y{\otimes }_k{X}^{\ast },M)\cong {\mathrm{Hom}}_A(Y,{\mathrm{Hom}}_{A^{\mathrm{o}\mathrm{p}}}(X^{\ast },M)),$ so $E_X^l$ is a right adjoint. Both auxiliary bimodules are finite-dimensional, and the displayed currying maps respect the outer actions, checked on elementary tensors. Transport through [F1] therefore makes evaluation on Rex a left adjoint and evaluation on Lex a right adjoint. By [F4], they preserve coends and ends respectively.

In the model of step 2.1, $N^r\cong A^{\ast }{\otimes }_A-$ and $N^l\cong {\mathrm{Hom}}_A(A^{\ast },-)$; the co-regular bimodule $A^{\ast }$ is in particular an $(A,A)$-bimodule, so [F9] applied to $M=A^{\ast }$ exhibits an adjunction $N^r⊣N^l$ with unit ${\eta }_X(x)(\lambda )=\lambda \otimes x$ and counit ${\epsilon }_Y(\lambda \otimes \phi )=\phi (\lambda )$, and the triangle identities hold by [F9] (Adjunction by unit, counit, and the triangle identities, $(S,R)$-bimodules and commuting left and right scalar actions, Linear map between vector spaces over the same field).

By [F3], ${\mathrm{\Psi }}^l(1_{\mathcal{A}})$ is the coend of $\bar{a}⊠a$ and ${\mathrm{\Psi }}^r(1_{\mathcal{A}})$ is its end. The equivalence ${\mathrm{\Phi }}^r$ preserves the first, and ${\mathrm{\Phi }}^l$ preserves the second, by [F4]. Applying the corresponding evaluation functors, which preserve these universal objects by step 3.1, gives pointwise universal objects in $\mathcal{A}$. Formula [F2] identifies their diagrams as ${\mathrm{Hom}}_{\mathcal{A}}(X,a{)}^{\ast }{\otimes }_{k}a$ and ${\mathrm{Hom}}_{\mathcal{A}}(a,X){\otimes }_{k}a$, respectively. Their universal maps are the evaluations of the images of the cowedges and wedges of [F3]. Consequently these are precisely the asserted coend formula for $N^r(X)$ and end formula for $N^l(X)$.

The formulas of step 4.1 refer only to the intrinsic data of $\mathcal{A}$: its hom functors, the finite-dimensional $k$-dual and the copowers of [F7]. For a second module model the same description therefore applies, and by the uniqueness of (co)ends [F8] the two resulting endofunctors are related by a unique compatible natural isomorphism; hence $N_{\mathcal{A}}^r$ and $N_{\mathcal{A}}^l$ are independent of the module model up to canonical natural isomorphism.

The identifications ${\mathrm{\Psi }}^r(1_{\mathcal{A}})\cong A$ and ${\mathrm{\Psi }}^l(1_{\mathcal{A}})\cong A^{\ast }$ of step 2.1 exhibit the regular and the co-regular bimodule as the images of the identity under ${\mathrm{\Psi }}^r$ and ${\mathrm{\Psi }}^l$; no isomorphism between these two bimodules, and no equivalence property of $N_{\mathcal{A}}^r$ or $N_{\mathcal{A}}^l$, is asserted or used, and the companion examples page records a finite category where they differ. Only the given finite-dimensional data and the finite (co)limits they determine are used, so no commutativity of $A$ and no further choice principle enter, and nothing is inferred from unrestricted completeness or cocompleteness.