Product of compact funcoids is compact
I have proved (for any products, including infinite products): Product of directly compact funcoids is directly compact. Product of reversely compact funcoids is reversely compact. Product of compact funcoids is compact. The proof is in my draft article and is not yet thoroughly checked for errors.
On the definition of compact funcoids
[I found that my computations below are erroneous, namely $latex \mathrm{Cor} \langle f^{-1}\rangle \mathcal{F} \neq \langle \mathrm{CoCompl} f^{-1}\rangle \mathcal{F}$ in general (the equality holds when $latex \mathcal{F}$ is a set).]