Unit in strict monoidal category

Question

Let $(\mathcal{C}, \otimes, 1)$ be a strict monoidal category.

If there exists an isomorphism between $1$ and an object $X$, does it follow that $1=X$? (if not, can you give a counterexample?).

Answer 1

Consider the category with two objects $A$ and $B$ such that any two objects have a unique morphism between them. Define $\otimes$ by $A \otimes A = A$, $A \otimes B = B$, $B \otimes A = B$ and $B \otimes B = A$. You can check that this is strictly unital and associative ($\otimes$ is addition mod 2). Since any two objects have a unique morphism between them, the functorial action of $\otimes$ is trivial. Nonetheless, $B$ is isomorphic but not equal to the unit $A$.

This example generalizes. Any monoid can be made into a strict monoidal category by using the indiscrete (or discrete) category on the underlying set of the monoid.