Example 12.18.2. Let \mathcal{A}, \mathcal{B}, \mathcal{C} be additive categories. Suppose that
is a functor which is bilinear on morphisms, see Categories, Definition 4.2.20 for the definition of \mathcal{A} \times \mathcal{B}. Given complexes X^\bullet of \mathcal{A} and Y^\bullet of \mathcal{B} we obtain a double complex
in \mathcal{C}. Here the first differential K^{p, q} \to K^{p + 1, q} is the morphism X^ p \otimes Y^ q \to X^{p + 1} \otimes Y^ q induced by the morphism X^ p \to X^{p + 1} and the identity on Y^ q. Similarly for the second differential.
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