A question regarding $Hom(X,X)$ (or $Mor(X,X)$)

Question

I refer to Rankeya's answer on this question.

Shouldn't $Mor(X,X)$ consist of monomorphisms only, for each morphism to have an inverse?

Answer 1

$\mathbf{Mor}(X,X)$ is always a monoid. Rankeya defines $\mathbf{Mor}(X,X)$ to be $G$, a group. $G$ has inverses by definition; hence, each member of $\mathbf{Mor}(X,X)$ has an inverse.