These are going to be a straight-to-the-point questions:
What is the difference between a free monoid and a Kleene Closure (Star) when generated by the set $A=\{1\}$?
Let $A^*$ be the free monoid over $A$ and $A^+$ be the Kleene Closure over $A$. If we categorize the set $A$ and the two algebraic structures $A^*$ and $A^+$, we define two functors $F^*:A\to A^*$ and $F^+:A\to A^+$. Can either functor become the equivalence of categories (e.g., $F^*:A\cong A^*$ and $F^+:A\cong A^+$)?
Thank you.
Kleene closure describes the submonoid of a free monoid generated by a subset of it. I don't understand the second half of your question.