Let $\mathcal E=\mathsf{Sh}(\mathsf C,J)$ be a Grothendieck topos. Suppose, for some representable sheaf $\mathbf{ay}C$, there are $H,G\in \Omega ^{\mathbf{ay}C}$ such that $H\cup G=\mathbf{ay}C$. I am having some trouble interpreting this statement with sheaf semantics. Below is a quick interpretation which I am unable to follow. I tried to add the details of what I can understand, and would appreciate some help filling in the rest.

[![enter image description here][1]][1]

First of all, since the Yoneda embedding is dense and sheafification commutes with colimits, I think we can may always assume generalized elements are representables. Hence $H,G\in \Omega ^{\mathbf{ay}C}$ correspond to a pair of arrows $\mathbf{ay}D\rightrightarrows \Omega ^{\mathbf{ay}C}$ in $\mathcal E$. Then, by the universal property of $\Omega$ and Cartesian closedness this amounts to two subobjects $$H\rightarrowtail \mathbf{ay}C\times \mathbf{ay}D,\; G\rightarrowtail \mathbf{ay}C\times \mathbf{ay}D,$$ whose union is $\mathbf{ay}C\times \mathbf{ay}D$. The union of these subobjects is $\mathbf{ay}C\times \mathbf{ay}D$.

That's the first three sentences, but the fourth one I'm confused about. This is probably because I don't understand the internal logic, what how is "a cover $E_\alpha\to C\times D$ possible? These should be arrows in the site $\mathsf C$, but the Yoneda embedding does not commute with products, so $\mathbf{ay}C\times \mathbf{ay}D\ncong \mathbf{ay}(C\times D)$ (I don't know whether this is even relevant)... I am confused here.

I know from Maclane and Moerdijk that $C\models \phi(\alpha)\vee \psi(\alpha)\iff$ there's a covering $(f_i:C_i\to C)$ such that for each index, $C_i\models \phi(\alpha f_i)\vee C_i\models \phi(\alpha f_i)$, but am confused about applying this.

How to understand the excerpt in the image?

  [1]: https://i.sstatic.net/wazCA.png