Consider the category of (proximally) continuous maps (entirely defined monovalued functions) between endo-funcoids.

Remind from my book that morphisms $latex f: A\rightarrow B$ of this category are defined by the formula $latex f\circ A\sqsubseteq B\circ f$ (here and below by abuse of notation I equate functions with corresponding principal funcoids).

Let $latex F_0, F_1$ are endofuncoids,

We define $latex F_0\times F_1 = \bigsqcup \left\{ \Phi \in \mathsf{FCD} \,|\,

\pi_0 \circ \Phi \sqsubseteq F_0 \circ \pi_0 \wedge \pi_1 \circ \Phi

\sqsubseteq F \circ \pi_1 \right\}$

(here $latex \pi_0$ and $latex \pi_1$ are cartesian projections).

**Conjecture** The above defines categorical direct product (in the above mentioned category, with products of morphisms the same as in **Set**).

This conjecture is probably the single most important conjecture in general topology. Please help me to solve it.

Earlier I conjectured that sub-atomic product of funcoids or displaced product of funcoids are categorical direct products. But the product introduced in this blog post is (in my opinion) the most important of all different products of funcoids, the candidate for “canonical” product of funcoids.

Again, I ask for help to solve this conjecture.

We can apply the same considerations also to reloids (and uniformly continuous maps between them).

I conjectured that $latex F_0\times F_1 = \bigsqcup \left\{ \Phi \in \mathsf{FCD} \,|\, \pi_0 \circ \Phi = F_0 \circ \pi_0 \wedge \pi_1 \circ \Phi = F \circ \pi_1 \right\}$.

This has turned out to be false, see

http://math.stackexchange.com/questions/480785/some-inequalities-about-binary-relations