Developing intuition about free groups / the universal property I'd like to check my intuition here since my chapter on free groups had my brain in a knot for a little while.
I know a free group is a group basically defined by all the reduced words of elements found in some indexing set, and that we can define homomorphisms from every free group into any group simply by defining where the generators of the free group are sent. 
Am I right in describing this intuition: The preimage of the homomorphism, in the free group, could be considered to describe the order we apply different elements (images of the generators of the free group) in the original group, without keeping track of where we end up. The image under the homomorphism would be the ending "location," effectively "resolving" the sequence of "actions" described by the element of the free group. 
So, in one example of this, I am thinking of the free group generated by moveup/movedown and moveleft/moveright, with rank 2, and then the same group action on the set of a 3x3 grid "Snake-Style" aka Z_3 x Z_3, and the homomorphism from the free group (Left - Down - Down - Down) would map to simply moving Left on the grid if you follow the mapping you'd expect. 
But I could have decided to have "moveup" map to "move up" and "move left" also map to "move up" and then left-down-down-down would, in the homomorphism, simply move me up once and then down 3 times, effectively moving me down two times (which may have resulted in a looparound).
Is this a reasonable understanding of the universal property?
 A: This is correct. For further intuition, think about the free group $F_2$ on two generators $a$ and $b$ as the set of all grid paths on $\mathbb{Z}\times\mathbb{Z}$ starting at $(0,0)$ and moving between horizontally or vertically adjacent vertices without immediate backtracking. The paths $(0,0)\to (0,1)\to(1,1)$ and $(0,0)\to (1,0)\to(1,1)$ are considered distinct. Paths are added similarly as vectors (with potential cancelling). Now we map $(F_2,\text{path addition})$ to $(\mathbb{Z}\times\mathbb{Z},+)$ by looking only at the endpoint of such a path. We loose the information on the exact journey of the path, i.e., the order in which the moves occurred, but we do know the net amount of horizontally and vertically moves. This is equivalent to introducing the relation $ab=ba$ into the free group. $\mathbb{Z}\times\mathbb{Z}$ has in fact the presentation $\left<a,b\;|\;ab=ba\right>.$
Think of the Cayley graph of $F_2$ as representing all of these paths, and think how this is mapped to the rectangular grid.
