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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?

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  • $\begingroup$ What you wrote doesn't seem to make sense. What is $X$? $\endgroup$ – Nex Oct 22 '15 at 8:35
  • $\begingroup$ Oh, sorry. X and Y are objects in Category C. $\endgroup$ – user282732 Oct 22 '15 at 8:37
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    $\begingroup$ Are you suggesting that an object $X$ has elements? $\endgroup$ – Nex Oct 22 '15 at 8:38
  • $\begingroup$ Yes. I considered that Sets as C. But I think I have to rewire it more generally... $\endgroup$ – user282732 Oct 22 '15 at 8:41
  • $\begingroup$ 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. $\endgroup$ – Nex Oct 22 '15 at 8:41
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Kan extensions might be what you're thinking of:

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)

I believe you can find the better part of a chapter devoted to them in CWM!

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