My question heads towards an intuition of universal properties. Take for example a group $G$ and a subset $M$. Then $F_M$, the free group on $M$ together with a map of sets $i:M\to U(F_M)$ where $U$ denotes the forgetful functor, is defined by the universal property that any map of sets $M\to U(H)$ factors through $U(F_M)$ for a group $H$. A concrete model for the free group is given by the free abelian group with basis $M$. Although I understand that universal properties fix objects up to isomorphism I am not sure about the following two things:
1) The explicit construction of $F_M$ clearly reveals it to be the smallest subgroup of $G$ containing $M$ in the sense that it is the group generated by $M$. How could this fact be concluded from the universal property?
2) More generally: Universal objects are often charaterized as being the smallest or most general objects regarding a certain property. I understand this in examples like the free group, the completion of a topological space, the tensor algebra, etc. However, I wonder if being the smallest or most general can be expressed (or even defined) in a precise categorical sense?
Concerning the last question: I know that any universal object is initial/terminal in a suitable factor category, however, I can not make sense of this...