Remark 13.10.9. Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ be additive categories and assume $\mathcal{C}$ has countable direct sums. Suppose that

is a functor which is bilinear on morphisms. This determines a functor

See Homology, Example 12.18.2.

For a fixed object $X^\bullet $ of $\text{Comp}(\mathcal{A})$ the functor

\[ K(\mathcal{B}) \longrightarrow K(\mathcal{C}), \quad Y^\bullet \longmapsto \text{Tot}(X^\bullet \otimes Y^\bullet ) \]is an exact functor of triangulated categories.

For a fixed object $Y^\bullet $ of $\text{Comp}(\mathcal{B})$ the functor

\[ K(\mathcal{A}) \longrightarrow K(\mathcal{C}), \quad X^\bullet \longmapsto \text{Tot}(X^\bullet \otimes Y^\bullet ) \]is an exact functor of triangulated categories.

This follows from Remark 13.10.8 since the functors $\text{Comp}(\mathcal{A}) \to \text{DoubleComp}(\mathcal{C})$, $Y^\bullet \mapsto X^\bullet \otimes Y^\bullet $ and $\text{Comp}(\mathcal{B}) \to \text{DoubleComp}(\mathcal{C})$, $X^\bullet \mapsto X^\bullet \otimes Y^\bullet $ are immediately seen to be compatible with homotopies and termwise split short exact sequences and hence induce exact functors of triangulated categories

Observe that for the first of the two the isomorphism

involves signs (this goes back to the signs chosen in Homology, Remark 12.18.5).

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