Find a suitable categorical product (related to proximity spaces)

Question

Funcoid is a binary relation $\delta$ between sets such that it holds for every $I$, $J$, $X$, $Y$ that:

1. $I\cup J\mathrel{\delta}Y\Leftrightarrow I\mathrel{\delta}Y\lor J\mathrel{\delta}Y$
2. $X\mathrel{\delta}I\cup J\Leftrightarrow X\mathrel{\delta}I\lor X\mathrel{\delta}J$
3. not $\emptyset\mathrel{\delta}Y$
4. not $X\mathrel{\delta}\emptyset$

You see, funcoids are a special case of proximity spaces.

A function $f$ is continuous from a funcoid $\delta_1$ to a funcoid $\delta_2$ when $X\mathrel{\delta_1}Y \Rightarrow f[X]\mathrel{\delta_1}f[Y]$ for all $X$, $Y$. ($f[X]$ denotes the image of the set $X$ by $f$.)

Evidently, if we take funcoids as objects and functions as morphisms, then we get a category.

Question: What is a (nontrivial) categorical product in this category? (I want something like the Tychonoff product of topological spaces, but for funcoids instead.)

Or what is the "standard" categorical product in the category on proximity spaces? (If we replace funcoids by proximity spaces in our category.)