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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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At a guess, I would say you're supposed to use the fact that functors which have a right adjoint preserve colimits. –  Zhen Lin Jun 5 '11 at 15:12
    
I don't understand what you are asking. The Freyd Adjoint Functor theorem tells you how to construct a left adjoint to the underlying set functor $\mathbf{G}\colon\mathscr{G}roup\to\mathscr{S}et$ (since $\mathbf{G}$ satisfies the required conditions). The left adjoint will respect colimits, which means it sends coproducts in $\mathscr{S}et$ to coproducts in $\mathscr{G}roup$. But how is it you want to "get" the free product of two arbitrary groups from this? It's not true that every coproduct in $\mathscr{G}roup$ is the image of a coproduct in $\mathscr{S}et$... –  Arturo Magidin Sep 8 '11 at 16:11
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up vote 3 down vote accepted

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.

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Corollary 3 in my copy (corrected sixth printing, 1994) is in page 209... –  Arturo Magidin Sep 8 '11 at 16:07
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