 If formula_101 and both formula_102 and formula_103 are finitely generated, then so is formula_17.The object formula_17 is finitely generated if, and only if, for any directed system formula_106 in formula_107 in which each morphism is a monomorphism, the natural morphism formula_108 is an isomorphism.A Grothendieck category need not contain any non-zero finitely generated objects.