Any object in a locally noetherian Grothendieck category has a noetherian subobject

Question

If $\mathcal{A}$ is a locally noetherian Grothendieck category, is that straightforward the fact that any object $M$ in $\mathcal{A}$ has a noetherian subobject?

Answer 1

Yes: Suppose $\{M_i\}_{i\in I}$ is a generating set of Noetherian objects in the given locally Noetherian Grothendieck category ${\mathscr A}$. Then for any nonzero $X\in{\mathscr A}$ there exists some $i\in I$ and a non-zero morphism $\varphi: M_i\to X$. The image of this morphism is a nonzero Noetherian subobject of $X$.

Even more: $X$ is the direct limit of the direct system of Noetherian subobjects.