The Stacks project

Proof. Choose a K-injective complex $\mathcal{I}^\bullet$ representing $M$, a K-injective complex $\mathcal{J}^\bullet$ representing $L$, and any complex of $\mathcal{O}_ X$-modules $\mathcal{K}^\bullet$ representing $K$. By Lemma 20.41.2 there is a map of complexes

The complexes of $\mathcal{O}_ X$-modules $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{J}^\bullet , \mathcal{I}^\bullet )$, $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{J}^\bullet )$, and $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{I}^\bullet )$ represent $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M)$, $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$, and $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, M)$. If we choose a K-flat complex $\mathcal{H}^\bullet$ and a quasi-isomorphism $\mathcal{H}^\bullet \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{J}^\bullet )$, then there is a map

whose source represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M) \otimes _{\mathcal{O}_ X}^\mathbf {L} R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$. Composing the two displayed arrows gives the desired map. We omit the proof that the construction is functorial. $\square$