Suppose we have a monoidal category $(\mathcal{C},\otimes)$. I am interested in the conditions or situations where the following do and do not hold:
For any objects A, B of $\mathcal{C}$, the induced map $\operatorname{End}(A) \times \operatorname{End}(B) \to \operatorname{End}(A \otimes B)$ is injective (by induced map, I mean the map induced by the bifunctoriality).
For any objects A, B of $\mathcal{C}$, the induced map $\operatorname{Aut}(A) \times \operatorname{Aut}(B) \to \operatorname{Aut}(A \otimes B)$ is injective (this is a restriction of the map in (1)).
(1) is stronger than (2). I think I have a proof that in the case that $\otimes$ is a product or coproduct (i.e., we have a Cartesian or co-Cartesian (?) monoidal category) [EDIT: My proof was wrong, as the counterexamples below show], both (1) and (2) hold, i.e., the maps on the two factors are "determined" by the map on the (co)-product. On the other hand, when $\otimes$ is the tensor product of modules over a commutative unital ring, (1) and (2) need not hold, though I think they do hold when the ring is a field.
So my question is: what are more examples where (1) and (2) do hold, what are more examples where they don't, and are there some other conditions/properties which would imply (1) and (2)?