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

$\otimes : \mathcal{A} \times \mathcal{B} \longrightarrow \mathcal{C}, \quad (X, Y) \longmapsto X \otimes Y$

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

$\text{Comp}(\mathcal{A}) \times \text{Comp}(\mathcal{B}) \longrightarrow \text{DoubleComp}(\mathcal{C}), \quad (X^\bullet , Y^\bullet ) \longmapsto X^\bullet \otimes Y^\bullet$

See Homology, Example 12.18.2.

1. 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.

2. 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

$K(\mathcal{B}) \to K_{first}(\text{DoubleComp}(\mathcal{C})) \quad \text{and}\quad K(\mathcal{A}) \to K_{second}(\text{DoubleComp}(\mathcal{C}))$

Observe that for the first of the two the isomorphism

$\text{Tot}(X^\bullet \otimes Y^\bullet [1]) \cong \text{Tot}(X^\bullet \otimes Y^\bullet )[1]$

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

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