Monoid and category

Question

Let $M$ be a monoid. Let $a, b$ be elements of $M$. We denote $\operatorname{Hom}(a, b) = \{s\ |\ sb = a\}$. Then we get a category whose set of objects is $M$. We denote this category by $C(M)$. Let $M, N$ be monoids. Suppose $C(M)$ is isomorphic to $C(N)$. Is $M$ isomorphic to $N$?

Answer 1

Your claim is false. Observe that, if $M$ is a group, then $C (M)$ is a category where there is exactly one morphism between any two objects. But the isomorphism class of such a category is determined uniquely by the cardinality of its object set. Thus, for example, $C (\mathbb{Z} / 4)$ is isomorphic to $C (\mathbb{Z} / 2 \times \mathbb{Z} / 2)$.