The Stacks project

Remark 15.71.2. Let $R$ be a ring. The category $\text{Comp}(R)$ of complexes of $R$-modules is a symmetric monoidal category with tensor product given by $\text{Tot}(- \otimes _ R -)$, see Lemma 15.58.1. Given $L^\bullet $ and $M^\bullet $ in $\text{Comp}(R)$ an element $f \in \mathop{\mathrm{Hom}}\nolimits ^0(L^\bullet , M^\bullet )$ defines a map of complexes $f : L^\bullet \to M^\bullet $ if and only if $\text{d}(f) = 0$. Hence Lemma 15.71.1 also tells us that

\[ \mathop{\mathrm{Mor}}\nolimits _{\text{Comp}(R)}(K^\bullet , \mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , M^\bullet )) = \mathop{\mathrm{Mor}}\nolimits _{\text{Comp}(R)}(\text{Tot}(K^\bullet \otimes _ R L^\bullet ), M^\bullet ) \]

functorially in $K^\bullet , L^\bullet , M^\bullet $ in $\text{Comp}(R)$. This means that $\mathop{\mathrm{Hom}}\nolimits ^\bullet ( - , -)$ is an internal hom for the symmetric monoidal category $\text{Comp}(R)$ as discussed in Categories, Remark 4.43.12.

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