I am reading Gortz's Algebraic geometry book, Vol.2 , p.207 and some qestion arises. I think that I am begginer of cohomology theory of schemes so please understand.
Let $f:X\to Y$ be a morphism of ringed spaces, where $Y=(*,A)$ possibly be a scheme such that the underliying space is consists of a single point and $A$ is the ring of global sections of $Y$. Let $\mathcal{F}$ and $\mathcal{G}$ be a complex of $\mathcal{O}_X$-modules. Then in the page p.207 the author defines cup product $ \cup : H^{p}(X,\mathcal{F}) \times H^{q}(X,\mathcal{G})\to H^{p+q}(X, \mathcal{F} \otimes_{\mathcal{O}_X}^{L} \mathcal{G})$ as follows : Fix cohomology classes
$$ \alpha \in H^{p}(X,\mathcal{F}) = \operatorname{Ext}^p_{\mathcal{O}_X}(\mathcal{O}_X,\mathcal{F})=\operatorname{Hom}_{D(X)}(\mathcal{O}_X[-p],\mathcal{F}),$$ $$ \beta\in H^{q}(X,\mathcal{G}) = \operatorname{Ext}^q_{\mathcal{O}_X}(\mathcal{O}_X,\mathcal{G})=\operatorname{Hom}_{D(X)}(\mathcal{O}_X[-q],\mathcal{G}) ,$$
where the equalities are given by $(21.4.4)$ and by our definition of the Ext modules. Using $\mathcal{O}_X[-p]\otimes^{L}\mathcal{O}_X[-q]=\mathcal{O}_X[-p-q]$ ( why? ), the functoriality of the derived tensor product yields an element
$$ \alpha \cup \beta \in \operatorname{Hom}_{D(X)}(\mathcal{O}_X[-p-q],\mathcal{F}\otimes_X^L\mathcal{G})=H^{p+q}(X,\mathcal{F}\otimes_X^L\mathcal{G}).$$
My question is, why some map $$ \operatorname{Hom}_{D(X)}(\mathcal{O}_X[-p],\mathcal{F}) \times \operatorname{Hom}_{D(X)}(\mathcal{O}_X[-q],\mathcal{G}) \to \operatorname{Hom}_{D(X)}(\mathcal{O}_X[-p]\otimes^L \mathcal{O}_X[-q],\mathcal{F}\otimes_X^L\mathcal{G}) $$ exists? What is exact meaning of the functoriality of the derived tensor product?