Let $(R/A)_\Delta$ be the prismatic site over $R$ relative to a prism $(A, I)$, then it is known that $(R/A)_\Delta$ admits finite non-empty coproduct, for instance, by Cor. 5.2 in Bhatt's lecture notes V on prismatic cohomology.

My question is, if we consider the absolute prismatic site $R_\Delta$, does $R_\Delta$ still always have finite non-empty coproduct?

If we restrict to the existence of finite non-empty self coproducts, it seems to me that the answer is yes, as it has been suggested (implicitly) in work:

Arthur-César Le Bras and Johannes Anschütz: Prismatic Dieudonné theory.

Or an upcoming work of Bhatt and Scholze, see the notes of Scholze on this topic in the RAMpAGe Seminar.

A more concrete question is the following:

Let $R=O_K$ be a complete discrete valuation ring of mixed characteristic with perfect residue field $k$, and let $(\mathfrak{S},(E))$ be a Breuil-Kisin prism in $R_\Delta$, i.e., $\mathfrak S=W(k)[\![u]\!]$ and $E \in W(k)[u]$ is the Eisenstein polynomial of a uniformizer of $O_K$. In the above two works, they claim the self product of $(\mathfrak{S},(E))$ in $R_\Delta$ exists and is equal to a prismatic envelop of $\mathfrak{S}\otimes_{W(k)}\mathfrak{S}$. Why is the tensor product over $W(k)$? I know for $(R/A)_\Delta$, the similar construction takes tensors over $A$, but it is clear that $(\mathfrak{S},(E))$ is not a prism over $(W(k),(p))$.