The Yoneda pairing, hypercohomology, and cup product

Let $$\mathcal{F}$$ and $$\mathcal{G}$$ be coherent analytic sheaves on $$\mathbb{P}^n$$. Let $$\mathcal{F}_\bullet$$ be a locally free resolution of $$\mathcal{F}$$. In Principles of Algebraic Geometry by Griffiths and Harris, $$\text{Ext}^p(\mathcal{F},\mathcal{G})$$ is defined as the hypercohomology of the complex $$\mathcal{Hom}(\mathcal{F}_\bullet,\mathcal{G})$$, i.e., the cohomology of the complex $$\bigoplus_{p=k+\ell} C^k(\mathfrak{U},\mathcal{Hom}(\mathcal{F}_\ell,\mathcal{G}))$$, see pages 705 and 446. Here $$C^\bullet(\mathfrak{U},\mathcal{Hom}(\mathcal{F}_\ell,\mathcal{G}))$$ denotes the Čech complex with respect to some affine open cover $$\mathfrak{U}$$ of $$\mathbb{P}^n$$.

If I understand correctly, the Yoneda pairing $$\text{Ext}^p(\mathcal{F},\mathcal{G}) \times \text{Ext}^q(\mathcal{G},\mathcal{H}) \rightarrow \text{Ext}^{p+q}(\mathcal{F},\mathcal{H})$$ should then be induced by the cup product in Čech cohomology. However, I fail to see precisely how this works out.

Edit: To clarify: What I am primarily interested in is how the Yoneda pairing can be expressed in terms of the concrete representatives when $$\text{Ext}$$ is realized as hypercohomology. In [HL, Section 10.1.1], there is an explicit cup product on hypercohomology, which then is said to induce a product $$Ext^i(F^\bullet,G^\bullet)\otimes Ext^j(E^\bullet,F^\bullet) \to Ext^{i+j}(E^\bullet,G^\bullet)$$. It is then also stated that "If we interpret $$Ext^i(E^\bullet,F^\bullet)$$ as $$Hom_{\mathcal D}(E^\bullet,F^\bullet[i])$$, where $$\mathcal{D}$$ is the derived category of quasi-coherent sheaves, then the cup product for Ext-groups is simply given by composition." To me, it is not clear neither how the cup product induces the product on $$\text{Ext}$$, nor why this product coincides with composition in the derived category. Any references to where this is discussed in more detail, or hints on how to prove this would be welcome.

[HL] Huybrechts, Lehn: The Geometry of Moduli Spaces of Sheaves

• I think these questions become clearer if you go full derived and see $\mathrm{Ext}^p(\mathcal{F},\mathcal{G})$ as the cohomology of the internal hom in the derived category of quasi-coherent analytic sheaves. – Denis Nardin Mar 11 at 11:34