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We show the Tychonoff's theorem for a Grothendieck category with a set of small projective generators. Strictly quasi-finite objects for semiartinian Grothendieck categories are characterized. We apply these results to the study of the Morita duality of dual algebra of a coalgebra

Topological linear compactness for Grothendieck categorires. Theorem of Thychonoff. Applications to coalgebras