> 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}} \\))?

>\[ \textrm{composing with } F : \mathbf{Set}^{\mathcal{C}} \to \mathbf{Set}^{\mathcal{D}} .\]

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}} \\))?