## § Kan Extensions: Key idea

• The key insight is to notice that when we map from $C \to E$ via $K$, then the $K(x)$ object that we get whose comma we form with $K \downarrow Kx$ also has an arrow $Kx \to Kx$ via the identity arrow. Thus we can think of $K \downarrow Kx$ as looking like ( -> Kx) -> Kx. So it's really the Kxin the  -> Kx that controls the situation.