The Stacks project

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.