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/algebraic-general-topology.htmlsee my book) several kinds of product of (any possibly infinite number) funcoids:

1.
cross-composition 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:

\\left\\langle f_{1}\\times f_{2}\\right\\rangle x=\\bigsqcup\\left\\{\\left\\langle f_{%1}\\right\\rangle X\\times^{\\mathsf{\\operatorname{FCD}}}\\left\\langle f_{2}\\right%\\rangle X\\hskip 10.0pt|\\hskip 10.0ptX\\in\\operatorname{atoms}x\\right\\}.

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 $\u$ are continuous (that is corresponding to a continuous funcoid) functions from the object of $\\mu$ to the object of $\u$ .
•
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.