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?
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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.