[FOM] Fraenkel-Mostowski-Specker method and category theory

[V. Yu. Shavrukov]

On Apr 3, 2006, at 18:18, solovay at math.berkeley.edu wrote:
>      My question is this: Are (a) and (b) the only reasons that two  
> such
> categories are equivalent. That is, if C(G_1, Gamma_1) and C(G_2,
> Gamma_2) are equivalent then are there subgroups H_1 of G_1 and H_2
> of G_2 [lying in the respective filters] such that letting Gamma_1'
> and Gamma_2' be the evident restricted filters we have C(H_1,
> Gamma_1') equivalent to C(H_2, Gamma_2').

I must be missing something about this question.  Isn't the answer  
automatically yes once we put  H_1 = G_1  and  H_2 = G_2 ?  I feel  
tempted to replace the conclusion by  (H_1, Gamma_1')  isomorphic to  
(H_2, Gamma_2').


regards,
Volodya Shavrukov