Eilenberg-Watts is a schematic equivalence of Hom categories

Let $\mathrm{\Phi }$ be the assignment $M\mapsto T_M$ on objects of the bimodule category and $f\mapsto (f\otimes 1_X)$ on morphisms. By [F1] the objects are additive cocontinuous functors and the morphisms are natural transformations, compatibly with identities and vertical composition, so $\mathrm{\Phi }$ respects the schematic category operations of [F2]; fixed Hom-collections are represented by sets by [F2].

$\mathrm{\Phi }$ is full and faithful: for every pair $M,M^{\mathrm{\prime }}$ the map ${\mathrm{\Phi }}_{M,M^{\mathrm{\prime }}}:{\mathrm{Hom}}_{B\text{-}A}(M,M^{\mathrm{\prime }})\to \mathrm{Nat}(T_M,T_{M^{\mathrm{\prime }}})$ is the bijection of [F3].

$\mathrm{\Phi }$ is split essentially surjective: by [F1] every additive cocontinuous $F$ is naturally isomorphic to $T_{F(A)}=\mathrm{\Phi }(F(A))$, and $F(A)$ together with that isomorphism is determined by $F$, so the required data are supplied without any selection.

The construction proving [F4] applies schematically: for $\eta :F\Rightarrow G$, define $E(\eta )$ as the unique bimodule map whose tensor transformation is ${\tau }_G^{-1}\circ \eta \circ {\tau }_F$, using the canonical comparisons ${\tau }_F:T_{F(A)}\Rightarrow F$ of [F1] and the bijection [F3]. Composition compatibility makes $E$ functorial, and $\tau$ is natural in $F$ by this defining equation. Since ${\tau }_{F,A}={\rho }_{F(A)}$, [F3] gives $E(\eta )={\eta }_A$. The tensor-unit maps give the other natural isomorphism $E(T_M)\cong M$: the reconstructed right action is $(m\otimes a)b=m\otimes ab$, and ${\rho }_M(m\otimes ab)=(ma)b$, while naturality follows on elementary tensors, so $E(F)=F(A)$ is a schematic quasi-inverse. No quantification over objects that are proper classes is needed.

Naturality in $M$ and $M^{\mathrm{\prime }}$: for a bimodule map $h:M_1\to M_2$ and $f^{\mathrm{\prime }}:M_2\to M^{\mathrm{\prime }}$, the bijection of [F3] sends $f^{\mathrm{\prime }}\circ h$ to the vertical composite of $(f^{\mathrm{\prime }}\otimes 1_X)$ with $(h\otimes 1_X)$ by the composition compatibility in [F3]; likewise postcomposition with a bimodule map corresponds to postcomposition with its tensor transformation. Hence the bijections are natural in both variables.

The bijection identifies natural isomorphisms with bimodule isomorphisms: if $\eta :T_M\Rightarrow T_{M^{\mathrm{\prime }}}$ is a natural isomorphism with associated $f$ and inverse ${\eta }^{-1}$ with associated $g$, then the identities ${\eta }^{-1}\circ \eta =1_{T_M}$ and $\eta \circ {\eta }^{-1}=1_{T_{M^{\mathrm{\prime }}}}$ translate under the compatibility of [F3] into $g\circ f=1_M$ and $f\circ g=1_{M^{\mathrm{\prime }}}$, because the bijection sends $1_{T_M}$ to $1_M$; conversely, for a bimodule isomorphism $f$ the transformation with components $f\otimes 1_X$ is a natural isomorphism with components $f^{-1}\otimes 1_X$. Hence $T_M\cong T_{M^{\mathrm{\prime }}}$ if and only if $M\cong M^{\mathrm{\prime }}$.

Steps 1.1-2.1 exhibit the equivalence $\mathrm{\Phi }$ with quasi-inverse $F\mapsto F(A)$, step 2.2 its naturality in both variables, and step 2.3 the isomorphism statement. No object or presentation was chosen, and no commutativity is used.