@Ken – I think they are related: given two monoids, a red one and a blue one, the smallest monoid containing both doesn't simply consist of red elements or blue ones, but also includes alternating sequences, eg red-blue-red, blue-red-blue-red.

In the same way, the join of red \\(\sim_P\\) and blue \\(\sim_Q\\) doesn't just consist of red links or blue links, but also includes alternating paths of links, eg red-blue-red, blue-red-blue-red.

I'm guessing there's some deep categorical reason for this, ie at some level the two constructions are basically the same.