Cool! That's even nicer than I thought. I was having a hard time seeing how the Kan extension was working so I thats why drew blue arrow from \\(H\\) to \\(Lan_{G}(H)\\) to help me realize that its just a function from \\(H\\) along \\(G\\). The picture makes much more sense now with the one-to-one and onto function from \\(\alpha\\) to \\(\beta\\).

![Left Kan Extension with one-to-one function](