  We solve the word problem for free double categories without equations
between generators by translating it to the word problem for 2-categories. This
yields a quadratic algorithm deciding the equality of diagrams in a free double
category. The translation is of interest in its own right since and can for
instance be used to reason about double categories with the language of
2-categories, sidestepping the pinwheel problem. It also shows that although
double categories are formally more general than 2-categories, they are not
actually more expressive, explaining the rarity of applications of this notion.
