Page "Stone–Čech compactification" Paragraph 20

The special property of the unit interval needed for this construction to work is that it is a cogenerator of the category of compact Hausdorff spaces: this means that if f and g are any two distinct maps from compact Hausdorff spaces A to B, then there is a map h from B to such that hf and hg are distinct.

Any other cogenerator ( or cogenerating set ) can be used in this construction.