Direct products in a category of funcoids
ADDED: I’ve proved that subatomic product^{} is the categorical product.
There are defined (\hrefhttp://www.mathematics21.org/algebraicgeneraltopology.htmlsee my book) several kinds of product of (any possibly infinite^{} number) funcoids:

1.
crosscomposition product

2.
subatomic product

3.
displaced product
There is one more kind of product, for which it is not proved that the product of funcoids are (pointfree) funcoids:
$$\u27e8{f}_{1}\times {f}_{2}\u27e9x=\bigsqcup \{\u27e8{f}_{1}\u27e9X{\times}^{\mathrm{FCD}}\u27e8{f}_{2}\u27e9X\mathit{\hspace{1em}}\mathit{\hspace{1em}}X\in \mathrm{atoms}x\}.$$ 
It is considered natural by analogy^{} with the category Top of topological spaces^{} to consider this category:

•
Objects are endofuncoids on small sets.

•
Morphisms between a endofuncoids $\mu $ and $\nu $ are continuous^{} (that is corresponding to a continuous funcoid) functions from the object of $\mu $ to the object of $\nu $.

•
Composition^{} is induced by composition of functions.
It is trivial to show that the above is really a category.
The product of functions is the same as in Set.
Title  Direct products in a category of funcoids 

Canonical name  DirectProductsInACategoryOfFuncoids 
Date of creation  20130912 19:40:35 
Last modified on  20130912 19:40:35 
Owner  porton (9363) 
Last modified by  porton (9363) 
Numerical id  11 
Author  porton (9363) 
Entry type  Definition 
Classification  msc 54J05 
Classification  msc 54A05 
Classification  msc 54D99 
Classification  msc 54E05 
Classification  msc 54E17 
Classification  msc 54E99 