We say that a finite, simple, undirected graph $G = (V,E)$ is *(product-)irreducible* if it is connected and there are no graphs $A,B$ such that $G\cong A\times B$ (where $\times$ denotes the categorical product of graphs).

Suppose that $G_i, H_j$ are irreducible graphs, where $i \in \{1,\ldots,n\}$ and $j\in\{1,\ldots,m\}$ for some positive integers $m,n\in\mathbb{N}$, and suppose that $$\prod_{i=1}^nG_i \cong \prod_{j=1}^m H_j.$$

**Questions.** Do we have $m=n$? And if yes, is there a bijection $\varphi:\{1,\ldots,n\}\to\{1,\ldots,n\}$ such that $G_i \cong H_{\varphi(i)}$ for all $i\in \{1,\ldots,n\}$?