Let $\mathcal{C}$ be a site on the category of schemes for some Grothendieck topology, $X$ a scheme, and let $F,G$ be two sheaves of abelian groups on $\mathcal{C}$. Do we have cup-product as follows? $$H^i(X,F)\otimes H^j(X,G)\rightarrow H^{i+j}(X,F\otimes_{\mathbb{Z}}G)$$ or even the relative version $$R^if_*F\otimes R^jf_*G\rightarrow R^{i+j}f_*(F\otimes_{\mathbb{Z}}G)$$ along a map $f:X\rightarrow Y$ of schemes.

For etale cohomology, it seems that Godement resolution is needed in the construction, see [Etale cohomology theory, the paragraph before Prop. 7,4,10] by Fu Lei, see also Appendix B of http://math.stanford.edu/~conrad/Weil2seminar/Notes/L12-13.pdf.

But in a general site (e.g. fppf site), we may not have Godement resolution. I checked the chapter "cohomology of sites" of Stack-Project, unfortunately, cup -product is in the list of topics that "should be discussed in this chapter, and have not yet been written"

It would be very helpful, if someone knows a reference.