# image of generator in filtered colimit in grothendieck category

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.

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