**Puzzle 110**

Since paths of length 0 in \\( \mathbf{Free}(G)\\) are the identity morphisms of that category and functors preserve identities, paths of length 0 from node \\(x\\) to itself must map to the identity function from the set \\(F(x)\\) to itself.

Since paths of length 0 in \\( \mathbf{Free}(G)\\) are the identity morphisms of that category and functors preserve identities, paths of length 0 from node \\(x\\) to itself must map to the identity function from the set \\(F(x)\\) to itself.