The following is one of a few (possibly non-equivalent) definitions of products of funcoids:
Definition Let 
for atomic reloids 
![a \mathrel{\left[ \prod^{[\mathcal{F}]} f \right]} b \Leftrightarrow \exists N \in \mathcal{F} \forall i \in N : \mathrm{Pr}^{\mathsf{RLD}}_i\, a \mathrel{[f_i]} \mathrm{Pr}^{\mathsf{RLD}}_i\, b.](https://s0.wp.com/latex.php?latex=a+%5Cmathrel%7B%5Cleft%5B+%5Cprod%5E%7B%5B%5Cmathcal%7BF%7D%5D%7D+f+%5Cright%5D%7D+b+%5CLeftrightarrow+%5Cexists+N+%5Cin+%5Cmathcal%7BF%7D+%5Cforall+i+%5Cin+N+%3A+%5Cmathrm%7BPr%7D%5E%7B%5Cmathsf%7BRLD%7D%7D_i%5C%2C+a+%5Cmathrel%7B%5Bf_i%5D%7D+%5Cmathrm%7BPr%7D%5E%7B%5Cmathsf%7BRLD%7D%7D_i%5C%2C+b.+&bg=ffffff&fg=000&s=0&c=20201002 "a \mathrel{\left[ \prod^{[\mathcal{F}]} f \right]} b \Leftrightarrow \exists N \in \mathcal{F} \forall i \in N : \mathrm{Pr}^{\mathsf{RLD}}_i\, a \mathrel{[f_i]} \mathrm{Pr}^{\mathsf{RLD}}_i\, b.")
Today I have proved that this really defines a funcoid. Currently the proof is present in draft of the second volume of my book,
A probably especially interesting case is if
This may possibly have some use in study of compact funcoids.