Showing MON and CAT are equivalent categories

Question

I am struggling with an equivalence of categories.
Let $\mathbf{Mon}$ be the category of monoids, and let $\mathbf{Cat}$ be the category whose objects are all categories with exactly one object. The morphisms from object $A$ to $B$ in $\mathbf{Cat}$ are all functors from $A$ to $B.$ I want to show that $\mathbf{Mon}$ and $\mathbf{Cat}$ are equivalent categories. Please help. Thanks in advance.

Answer 1

A monoid $(M,*,1)$ can be seen as a category with only one object $\bullet$, the same way a group is a groupoid with a single object. The composition of morphisms corresponds to the multiplication $"*"$ of elements and the rule $f\circ(g∘h)=(f∘g)∘h$ means that multiplication is associative, while the single identity arrow $1_\bullet$ is the unit of $*$ since $1∘f=f∘1=f$ for any $f\in M$.

A functor $F:M\to N$ between two categories with a single object is uniquely determined by its arrow function, and since there is only one hom-set, any function is allowed, as long as it respect the composition of arrows and the identity, i.e. $F(1_∙)=1_∙$ and $F(g∘f)=F(g)∘F(f)$. But these are just the properties that we require from a homomorphism between monoids. Hence, a functor between two monoids, considered as categories with a single object, is basically the same as a homomorphism, and vice versa.