# Why can't we define the naturals transformation between C->D functor and C->E functor

In CWM (and other text books), natural transformations are defined between functors $F : C \to D$ and $G : C \to D$, that is to say, functors whose destinations are the same Category $D$.

Why do we need this limit? Can't we define the natural transformations between functors $F : C \to D$ and $G : C \to E$?

Without the limit of destination, I think we can define the natural transformations as the functor $\eta : D \to E$ like:

Assume:

$C$ : Subcategory of $Sets$ so objects of $C$ are sets, and arrows of $C$ are functions.

(But I would like to remove this assumption)

$F : C \to D$ (functor)

$G : C \to E$ (functor)

$X,Y \in Obj_C$

Then:

Natural transformation is a functor $\eta : D \to E$, such that:

for any Arrow $f : X \to Y$ in Category $C$,

$\eta((F(f))(F(x))) = (G(f))(\eta(F(x)))$ for any $x \in X$

To define the equation in the last line, I assumed that $X$ has elements. Is there any way to remove this assumption?

• What you wrote doesn't seem to make sense. What is $X$? – Nex Oct 22 '15 at 8:35
• Oh, sorry. X and Y are objects in Category C. – user282732 Oct 22 '15 at 8:37
• Are you suggesting that an object $X$ has elements? – Nex Oct 22 '15 at 8:38
• Yes. I considered that Sets as C. But I think I have to rewire it more generally... – user282732 Oct 22 '15 at 8:41
• If $X$ is an object and $F$ is a functor, then $F(X)$ is an object and hence $F(f)F(X)$ does not make sense. – Nex Oct 22 '15 at 8:41

Given two functors $F:\mathscr{C} \rightarrow \mathscr{D}$ and $G:\mathscr{C} \rightarrow \mathscr{E},\space$ a left Kan extension G along F is a functor $Lan _{F}\space G:\mathscr{D} \rightarrow \mathscr{E}$ together with a natural transformation $\eta:G\Rightarrow Lan _{F}\space G \cdot F$ such that for any other pair $( H:\mathscr{D} \rightarrow \mathscr{E},\epsilon:G\Rightarrow HF)$, $\space \epsilon \space$ factors uniquely through $\eta. \space$ That is, $\space \exists! \alpha:Lan _{F}\space G\Rightarrow H\space$ such that $\space \epsilon= \alpha _{F} \cdot \eta . \space$(Riehl; CHT: Definition 1.1.1)