Let us take the following assumptions: $\mathscr{M}$ a monoidal category, $X,Y,Z$ three objects in the category, and $f: Y \to Z$ a morphism. If the morphism $$ \mathrm{id}_X \otimes f: X \otimes Y \to X \otimes Z $$ is an isomorphism then can we imply that $f$ is also an isomorphism?