Why is el(-)=$\int(-)$ a functor from functors to a slice cateory? I am reading up a little on category theory. I am trying to solve Riehl's problem (Category Theory in Context) 2.4.vii on page 72. I believe that it should be quite easy, but it appears to me that there must be some mistake in the question. I am using the notation used in the book, i.e. $\int\!F$ is the category of elements of the functor $F:C \to \text{Set}$ and $C/c$ is the category of slices containing maps $x \to c$ as objects and morphisms are commutative triangles. $F \downarrow G$ is the comma category of two functors with common codomain.
The exercise is: 
Show that the construction of the category of elements defines the action of a functor $\int\!(-):\text{Set}^C \to \text{CAT}/C$.
I guess one idea is to write $\int F \simeq y \downarrow F$ and show that $y \downarrow(-)$ is functorial. But I cannot see how $\text{CAT}/C$ could be the codomain of such a functor. Also I think, I want $C$ to be locally small, so that we can use Yonedas Lemma.
My problem is: Objects in $y \downarrow F$ (where $F$ is understood as $F:1\!\!1 \to \text{Set}^C$) are triples $(c,1\!\!1,\alpha:C(c,-) \Rightarrow F)$, i.e. a natural transformation $C(c,-) \Rightarrow F$. But the objects in $\text{CAT}/C$ are functors $G:\,? \to C$. Is there some identification (via Yoneda) going on which I did not see? Thank you!
 A: Thank you for this hint! I think, what you mean is: $\int$ maps a functor $F$ to the projection $\Pi:\int\!F \to C$ that maps $(c,x \in Fc) \mapsto c$. A natural transformation $\alpha:F \Rightarrow G$ is mapped to $\int\!\alpha^\ast$, that maps from $\Pi:\int\!F \Rightarrow \Pi:\int\!G$ by $\Pi(c,x \in Fc) \mapsto \Pi(c,\alpha_c(x) \in Gc)$. This is then functorial by the way the vertical composition of natural transformation is defined. This makes sense to me now.
I have one more question: The exercise asks to conclude that $F,G:\text{Set} \to C$ are natural isomorphic if $\int\!F \simeq \int\!G$. But the map that is defined in this way is not $\int$ itself but rather a projection on it. I would prefer to define $\int\!-$ as $\int\!-:\text{Set}^C \to \text{CAT}$, with $\alpha:F \Rightarrow G$ being mapped to $\int\alpha:(c,x) \mapsto (c,\alpha_c(x))$. I agree that the proof is pretty much the same. But in a sense this definition of the functorial action of $\int$ makes $\int\!F \neq \int\!$ (one being the category of objects and the other being the canonical projection from the category of objects to $C$). Why is the exercise written in this way? Thank you again!
