Let $(\mathcal{C}, \otimes , 1)$ be a monoidal category, and let $\mathrm{End}_{\mathcal{C}} (1)$ be the ground monoid of $\mathcal{C}$ - which is a commutative monoid. If $r_X : X \otimes 1 \to X$ denotes the right unit constrain of $\mathcal{C}$, then $f \cdot \alpha := r_Y (f \otimes \alpha) r_X^{-1}$ defines a right action on $\hom_{\mathcal{C}} (X;Y)$.

Question 1. When is this action free? That is, under what conditions $f \cdot \alpha = f$ implies $\alpha = \mathrm{id}_1$?

Question 2. Is it possible to loosen the conditions on $\hom_{\mathcal{C}} (1;Y)$?

For instance, in the category of $k$-modules this action is not free even when $k$ is a field (see Maxime Ramzi's comment). When $k$ is a ring the situation is even worse because of torsion.

Edit. More concretely, I'm trying to give sufficient conditions to mimic the following situation in the category of $k$-modules ($k$ a field): if $0 \neq v \in V$ and $\lambda \in k$, then $v = \lambda v$ implies $\lambda =1$.