Let G and H are two groups,I know that the coproduct of them is the free product,but how to get it from the adjoint functor theorem? And I also want to see some applications of the adjoint functor theorem,who can help me?
Depending on your definition of "getting the free product" (i.e. do you simply require it's mere existence with its categorical properties, or a more concrete description, which necessarily needs more "grubby" work ...), I suggest you simply look at Saunders Mac Lane's "Categories for the Working Mathematician" (2. ed., Springer GTM 5) Chapter IX. Special Limits, section 1. Filtered Limits, Corollary 3 on p. 213. Similar ideas lead to arbitrary small (co-)products for categories of arbitrary algebrae (in the sense of Universal Algebra; the main point is the solution set condition). Kind regards - Stephan F. Kroneck.