PlanetPhysics/Grothendieck Category Lemma

Introduction: proper generator
Let us recall that a generator of a Grothendieck category $$\mathcal{G}$$ is called proper if $$U$$ has the property that a monomorphism $$i: U' \to U$$ induces an isomorphism $$Hom_{\mathcal{G}}(U,U) \cong Hom_{\mathcal{G}}(U',U)$$ if and only if $$i$$ is an isomorphism (viz. p. 251 in ref. ).

Grothendieck category lemma
\begin{lemma} Any Grothendieck category $$\mathcal{G}$$ has a proper generator. \end{lemma}