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 free when $k$ is a field, but for a general ring there is torsion and this doesn't have to be true.