# The (candidate) construction of direct product in the category of continuous maps between endo-funcoids

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

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

Let $F_0, F_1$ are endofuncoids,

We define $F_0\times F_1 = \bigsqcup \left\{ \Phi \in \mathsf{FCD} \,|\, \pi_0 \circ \Phi \sqsubseteq F_0 \circ \pi_0 \wedge \pi_1 \circ \Phi \sqsubseteq F \circ \pi_1 \right\}$

(here $\pi_0$ and $\pi_1$ 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.