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.

References, comments, hints are much welcome.