Definition of natural transformations

Question

I am trying to understand Yoneda's lemma, and am trying to piece together the definition of a natural transformation. Wikipedia says that given two functors $F,G$ that map categories $C$ to $D$, a natural transformation is a family of morphisms such that given $X$ in $C$ there is some morphism in this family $\eta_X:F(X)\to G(X)$.

To me, this implies that a natural transformation is a family of morphisms that map objects from the image of $F$ to the image of $G$, rather than the commonly stated notion of mapping functors themselves. Is mapping functors themselves just a shorthand for this?

In the wikipedia article for the Yoneda lemma they write $$Hom(Hom(A,-),F)$$ but to me this implies a morphism between one functor to another, rather than a morphism between objects. Does the above question clarify this?

Answer 1

You are right that a natural transformation from $F$ to $G$ is a family of morphisms $F(X) \to G(X)$ that satisfies some compatibility conditions. However, one can define the category $\operatorname{Fun}(C,D)$ of functors $C \to D$. The morphisms $F \to G$ in this category are defined to be the natural transformations. This is why natural transformations are also said to be arrows between functors or similar things.

Answer 2

The correct notion of natural transformation is the one you give at the beginning of the question, i.e. "given two functors $F,G$ that map categories $C$ to $D$, a natural transformation is a family of morphisms in $D$ such that given $X$ in $C$, there is some morphism in this family $η_{X}:F(X)→G(X)$".

This is the usual notion of mapping between functors. I think you have misunderstood the concept of "family of morphisms between one functor to another". A mapping between functors does not mean that there is a function from the functor $F$ to the functor $G$ (if you think about this, it makes no sense to talk about a function between functors) but is defined just as above.