On linear functors on the abelian category of finitely generated modules over a Noetherian ring

Question

Let $R$ be a commutative Noetherian ring, and let $\mod R$ denote the abelian category of finitely generated $R$-modules. Let $F:\mod R \to \mod R$ be an $R$-linear functor (https://ncatlab.org/nlab/show/linear+functor). If $\phi: X \to Y$ is an $R$-linear map of finitely generated $R$-modules such that $\phi(X)\subseteq IY$ for some ideal $I$ of $R$, then is it true that $F(\phi)(F(X))\subseteq IF(Y)$?

Answer 1

Not necessarily.

For example, take $R=\mathbb{Z}$ and $I=2\mathbb{Z}$, with $F=\operatorname{Hom}(\mathbb{Z}/2\mathbb{Z},-)$, and $\phi:\mathbb{Z}/2\mathbb{Z}\to\mathbb{Z}/4\mathbb{Z}$ the injective homomorphism.