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?

Edit: Due to Daniel's comment below this does not seem to be true, so I wonder if there are some special type of categories where such problems do not arise. For example if the category is also abelian and the tensor product is linear in the obvious sense? Indeed if the category admits duals then this should be true?