# A conjecture about direct product of funcoids

I am attempting to define direct products in the category cont(mepfFcd) (the category of monovalued, entirely defined continuous pointfree funcoids), see this draft article for a definition of this category.

A direct product of objects may possibly be defined as the cross-composition product (see this article).

A candidate for product of morphisms $f_1:\mathfrak{A}\rightarrow\mathfrak{B}$ and $f_2:\mathfrak{A}\rightarrow\mathfrak{C}$ of this category is the pointfree funcoid $f_1\times^{(D)} f_2$ defined by the formula:

$\left\langle f_1 \times^{\left( D \right)} f_2 \right\rangle x = \bigcup \left\{ \langle f_1\rangle X \times^{\mathsf{FCD}} \langle f_2\rangle X \hspace{1em} | \hspace{1em} X \in \mathrm{atoms}^{\mathfrak{A}} x \right\}$.

The trouble is that I failed to prove that such a pointfree funcoid exists. So a conjecture:

Conjecture If $f_1$ and $f_2$ are monovalued, entirely defined funcoids then the pointfree funcoid defined by the above formula exists.