Suppose $\mathscr{A}$ is a grothendieck abelian category with generator $R$, is it true that $$\varinjlim \mathrm{Hom}(R,M_i) =\mathrm{Hom}(R,\varinjlim M_i)$$ if $M_i$ is a filtered system of objects.

  • $\begingroup$ In general, the functor $\operatorname{Hom}(R,-)$ preserves filtered colimits when $R$ is a finitely presented object: see here. $\endgroup$ – Fabio Lucchini Feb 6 at 14:27

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