I’ve discovered a new kind of product of funcoids, which I call subatomic product.

Definition Let $f : A_0 \rightarrow A_1$ and $g : B_0 \rightarrow B_1$ are funcoids. Then $f \times^{\left( A \right)} g$ (subatomic product) is a funcoid $A_0 \times B_0 \rightarrow A_1 \times B_1$ such that for every $a \in \mathrm{atoms}\,1^{\mathfrak{F} \left( A_0 \times B_0 \right)}$, $b \in \mathrm{atoms}\,1^{\mathfrak{F} \left( A_1 \times B_1 \right)}$

$a \mathrel{\left[ f \times^{\left( A \right)} g \right]} b \Leftrightarrow \mathrm{dom}\,a \mathrel{\left[ f \right]} \mathrm{dom}\,b \wedge \mathrm{im}\,a \mathrel{\left[ g \right]} \mathrm{im}\,b.$

This (subatomic) composition has the merit that for funcoids $f : A \rightarrow B$ and $g : A \rightarrow C$ the destination of product is $B \times C$ is the same as for categorical product in the category $\boldsymbol{\mathrm{Set}}$.

See This online draft article for details. There it is also proved that subatomic product exists.