Closed subcategories of a quotient category X/Y correspond bijectively to Y-closed subcategories of the base category, and closed subcategories of any category with compact projective generators correspond to ideals in those generators.
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Closed subcategories of quotient categories
Closed subcategories of a quotient category X/Y correspond bijectively to Y-closed subcategories of the base category, and closed subcategories of any category with compact projective generators correspond to ideals in those generators.