> But we've just seen that composing with \$$G: \mathcal{D} \to \mathcal{C}\$$ takes functors \$$F : \mathcal{C} \to \mathbf{Set} \$$ and gives functors \$$F \circ G : \mathcal{D} \to \mathbf{Set} \$$. In fact - this takes some work to check - this gives a functor
>$\textrm{composing with } F : \mathbf{Set}^{\mathcal{C}} \to \mathbf{Set}^{\mathcal{D}} .$

Shouldn't it be "\$$\textrm{composing with } G \$$"? After all this functor takes *any* functor \$$X \in \mathbf{Set}^{\mathcal{C}} \$$ to a functor in \$$\mathbf{Set}^{\mathcal{D}} \$$, not just the functor \$$F\$$.

Likewise, later, shouldn't we talk about "the left/right Kan extension along G" (of functors \$$F \in \mathbf{Set}^{\mathcal{C}} \$$)?