John

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](http://aether.co.kr/images/left_kan_extension_2.svg)