Puzzle Keith 50.1: What if G was an projection of Italians instead of Germans? How would \\(Lan_G(H)\\) behave then? That is let G be the unique functor such that G(Italians)=Italians.

By the definition of Lan, forall H, there's a natural one-to-one correspondence between

\[ \textrm{ morphisms from } \textrm{Lan}_G(H) \textrm{ to } F \]

in the category \\(\mathbf{Set}^\mathcal{C}\\) and

\[ \textrm{morphisms from } H \textrm{ to } F \circ G \]

in the category \\(\mathbf{Set}^\mathcal{D}\\).

So let's say H maps "Italians" to {Alessandro, Martina, Bianca}.

So then morphisms from \\( H \textrm{ to } F \circ G \\) are functions from {Alessandro, Martina, Bianca} to {Giuseppe, Giulia, Gian-Carlo, Alessandro, Martina, Bianca}. Just looking at cardinalities there are 6^3 of those, so there must also be 6^3 in morphisms from \\( \textrm{Lan}_G(H) \textrm{ to } F \\) also. So the mapping from the germans must be determanied..but I think that forces the set of Germans to be the empty set in \\( \textrm{Lan}_G(H) \\)