Derived tensor composition and the enhancement boundary

Let $k$ be a commutative ring and $A,B,C$ graded $k$-algebras. For bounded cochain complexes $F$ of graded $(B,A)$-bimodules and $G$ of graded $(A,C)$-bimodules satisfying the projectivity and boundedness hypotheses of the bounded-complex page, composition of the derived tensor functors corresponds to the degreewise balanced tensor product with the signed cochain totalization of Bounded graded bimodule complexes and signed tensor totalization. Internal degrees enter only the grading of the total complex, so no additional internal-degree sign is introduced, and the cochain sign depends only on the cochain degree: this is the convention of the internal shift $M\{r{\}}_d=M_{d-r}$ of Associative graded algebras, bimodules, and internal shifts, under which a shifted complex has the same differential and the same elements, unlike the cochain shift $[1]$ which has $X[1{]}^n=X^{n+1}$ and differential $-d_X^{n+1}$, so it lowers cochain placement by one.

The associativity, unit and cone-compatibility statements and the derived-tensor equivalences supplied by inverse complexes are exactly those of Bounded bimodule tensor is associative, unital, and compatible with cones and Supplied inverse bimodule complexes give derived tensor equivalences, applied with the graded balanced associators and unitors of Graded associativity, units, and internal-shift tensor isomorphisms; their projectivity, boundedness, homotopy and graded/cochain hypotheses are preserved verbatim, with each coherence identity an identity of underlying graded bimodules checked on elementary tensors.

This remark asserts only that supplied inverse complexes give those equivalences. It makes no assertion that an arbitrary abstract triangulated functor or natural transformation between derived categories is induced by a bimodule complex: a dg or stable enhancement with an appropriate notion of morphism would be needed for such a classification, and it lies outside this A/B pair. Likewise relative tensor categories, Radford's $S^4$ theorem, arbitrary Grothendieck categories and schemes are not prerequisites of this pair, and the internal shift $\{r\}$ of the graded theorem is the graded-module shift of Associative graded algebras, bimodules, and internal shifts, not the cochain shift $[1]$ of the bounded-complex page. No commutativity beyond $k$ and no choice are used.