I am reading these notes, and am trying to understand the reasoning in the last sentence of Theorem 13.18. The author shows that if $E$ and $E'$ are isogenous elliptic curves, then $\operatorname{End}^0(E')$ is isomorphic to a $\mathbb{Q}$-subalgebra of $\operatorname{End}^0(E)$, and vice versa, then concludes htat $\operatorname{End}^0(E') \cong \operatorname{End}^0(E)$. This appears to be an invokation of the Cantor-Bernstein-Schroeder (CBS) property, which is discussed at length here, but does not hold in all categories (various counterexamples are available around the internet). My question is - How do we know that the CBS property holds for finitely-generated $\mathbb{Q}$-algebras?
Or perhaps we are using something specific about the case of endomorphism algebras of elliptic curves, like the classification that immediately precedes this Theorem?