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.

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?

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.

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?

Added. This comes from page 149 of the paper Local Concepts in Synthetic Differential Geometry and Germ Representability by Bunge and Dubuc.

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.

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?

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.

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?