This is a reference request question. But let's start with a few definitions.
Let $L$ and $M$ be two bounded lattices. A multioperator $p$ for $L$ and $M$ is an application $$p : L \to Ft(M)^{op}$$ where $Ft(M)^{op}$ denotes the opposite lattice of filters over $M$ such that $p(x\vee y) = p(x) \vee p(y)$ and $p(0)=0$. (Since it is the opposite lattice, we have actually $p(0)=M$ and $p(x \vee y) = p(x) \cap p(y)$.)
A multimodal algebra is a boolean algebra $B$ with a multioperator $p_B$ for $B$ and $B$. We say that $f :(A,p_A)\to (B,p_B)$ is a morphism of multimodal algebra if it is a morphism of boolean algebra and if $$p_B(f(a)) = (f(p_A(a)))\uparrow$$ for all $a\in A.$ We note $MMA$ the category of multimodal algebra.
A multimodal space is a boolean space $X$ equiped with a binary closed relation $R_X$. We say that $f :(X,R_X)\to (Y,R_Y)$ is a morphism of multimodal space if it continuous and if $$f(R_X(-,x))=R_Y(-,f(x))$$ for all $x\in X$. We note $MMS$ the category of multimodal spaces.
Now my question is the following :
Is there any paper which established a dual equivalence between MMA and MMS ?
I know this a well-known result for the specialists, but it is surprisingly difficult to find a relevant source.