Why this intuition about natural transformations corresponds to its formal definition?

Almost everywhere people introduce the notion of natural transformations between two functors $F$, $G$ : $\textbf C \Rightarrow \textbf D$ by examples like what follows:

This is the intuition they approach with:

• Consider for example, the functors $(- \times B) \times C$ and $- \times ( B \times C): \textbf C \Rightarrow \textbf C$. For an object $\mathcal A$ we have the isomorphism $(A \times B) \times C \cong^{h_A} A \times ( B \times C)$ via $h_ A$. Now if we change $A$ into $\mathcal A'$, we still have a similar isomorphism $( A' \times B) \times C \cong^{h_{A'}} A' \times ( B \times C)$ via some morphism $h_ {A'}$. That is, the isomorphism holds independent of the choice of $A$. That is the idea of a natural transformation.

Now as they say, this can be formalized by the fact that:

• For any morphism $f: A \to A'$, the following diagram commutes:

Now, I don't get how that "substitution/change and holding a similar isomorphism" is taken from "existing a morphism $f:A \to A'$ and commuting the diagram"!

HINT: I'm not concerned about the so called isomorphism between those objects; let it be a simple morphism (as in the general case of a natural transformation) if you want.

Thanks.

The formalization of

• Consider for example, the functors $$(- \times B) \times C$$ and $$- \times ( B \times C): \textbf C \Rightarrow \textbf C$$. For an object $$\mathcal A$$ we have the isomorphism $$(A \times B) \times C \cong^{h_A} A \times ( B \times C)$$ via $$h_ A$$. Now if we change $$A$$ into $$\mathcal A'$$, we still have a similar isomorphism $$( A' \times B) \times C \cong^{h_{A'}} A' \times ( B \times C)$$ via some morphism $$h_ {A'}$$. That is, the isomorphism holds independent of the choice of $$A$$. That is the idea of a natural transformation.

is not a natural transformation, but merely the family of (iso)morphisms $$(A\times B)\times C\cong^{h_A}A\times(B\times C)$$. Notice that as a family, the morphisms $$h_A$$ are indexed by the objects $$A$$ of the category. More generally, the above intuition is only enough to explain why given two functors $$F,G\colon\mathbf C\to\mathbf D$$, we would want a family of morphisms $$FX\xrightarrow{h_X}GX$$ indexed by the objects $$X$$ of $$\mathbf C$$.

What is missing from the above intuition is exactly the naturality of the natural transformation, that is, the requirement that the family of morphisms be in addition compatible with the morphisms of $$\mathbf C$$ in an appropriate way. Thus I would say that the above intuition does not really capture the essential feature of a natural transformation, namely its naturality.

Now, intuition is necessarily a personal tool, so I can only give you my own thoughts, which are two.

First, consider not categories in general, but locally small categories. That is, consider a category $$\mathbf C$$ such that for each pair of objects $$X$$ and $$Y$$ the associated collection of morphisms is the collection of elements of a set $$Hom_{\mathbf C}(X,Y)$$, and such that the composition relation between morphisms is captured by a set function $$Hom(Y,Z)_{\mathbf C}\times Hom(X,Y)_{\mathbf C}\xrightarrow{\circ}Hom_{\mathbf C}(X,Z)$$.

Then each object $$X$$ determines an actual functor $$Hom_{\mathbf C}(X,-)\colon\mathbf C\to\mathbf Set$$ sending each object $$Y$$ of $$\mathbf C$$ to the set $$Hom_{\mathbf C}(X,Y)$$ and each morphism $$Y_1\xrightarrow{h}Y_2$$ in $$\mathbf C$$ to the post-composition function between sets $$Hom_{\mathbf C}(X,Y_1)\xrightarrow{Hom_{\mathbf C}(X,h)}Hom_{\mathbf C}(X,Y_2)$$ sending a morphism $$X\xrightarrow{g}Y_1$$ to its post-composition $$X\xrightarrow{g}Y_1\xrightarrow{g}Y_2$$ with $$h$$. In other words, $$h_*^X(h)=h\circ g$$. Such functors are called representable.

A natural question arises, which is this: what relationship between the functors $$Hom_{\mathbf C}(X_2,-)$$ and $$Hom_{\mathbf C}(X_1,-)$$ is induced by a morphism $$X_2\xrightarrow{f}X_1$$?

The answer is that any morphism $$X_2\xrightarrow{f} X_1$$ induces a pre-composition function of sets $$Hom_{\mathbf C}(X_2,Y)\xleftarrow{f^*_Y}Hom_{\mathbf C}(X_1,Y)$$, one for each object $$Y$$ in $$\mathbf C$$. So in particular we have a family of functions of sets $$f^*_Y$$ indexed by the objects $$Y$$ of $$\mathbf C$$, just as your paragraph suggests. But this family is also compatible in a very specific way with the morphisms in $$\mathbf C$$. Namely, associativity of composition $$(f\circ g)\circ h=f\circ (g\circ h)$$ is equivalently expressed by the statement that the diagram $$\require{AMScd} \begin{CD} Hom_{\mathbf C}(X_2,Y_1) @>{Hom_{\mathbf C}(X_2,h)}>> Hom_{\mathbf C}(X_2,Y_2)\\ @V{f^*_{Y_2}}VV @V{f^*_{Y_1}}VV \\ Hom_{\mathbf C}(X_1,Y_1) @>{Hom_{\mathbf C}(X_2,h)}>> Hom_{\mathbf C}(X_1,Y_2) \end{CD}$$ commutes. In this case, naturality of the family of morphisms amounts exactly to associativity. (If one takes a further small step in this direction and asks what are the natural transformations from $$Hom_{\mathbf C}(X,-)$$ to any functor $$\mathbf C\to\mathbf Set$$, you would discover the Yoneda lemma, and in particular that the opposite category $$\mathbf C^{op}$$ of $$\mathbf C$$ is exactly the category of representable functors and natural transformations between them).

A second way to intuitively grasp the naturality of natural transformations is to view a functor $$\mathbf C\xrightarrow{F}\mathbf D$$ as a structured family of objects of $$\mathbf D$$ indexed by the objects of $$\mathbf C$$, the structure specified by the family of morphisms of $$\mathbf D$$, indexed by the morphisms of $$\mathbf C$$, between the objects in the structured family of objects of $$\mathbf D$$. From this point of view, a natural transformation between two functors is a homomorphism between the two structured families of objects. The most common case of this are algebraically structured sets, e.g. groups. A set with a binary operation $$M$$ can be thought of as the family of sets $$\{e\},M,M\times M,M\times M\times M,\dots$$ with structure given by the iterated binary multiplications: $$M\times M\xrightarrow{\mu} M$$ sending $$(m_1,m_2)\mapsto m_1\cdot m_2$$, $$M\times M\times M\to M$$ sending $$(m_1,m_2,m_3)$$ to $$(m_1\cdot m_2)\cdot m_3$$ or $$m_1\cdot(m_2\cdot m_3)$$, etc. Then a set function $$M_1\xrightarrow{f} M_2$$ determines a family of morphisms $$M_1^n\xrightarrow{(f,f,\dots,f)}M_2^n$$, and naturality of that family will be exactly the statement that $$f$$ is a homomorphism of the binary operation.

• I think you've given the intuition slightly short shrift-you seem to ignore the idea that it should be somehow the same isomorphism no matter which objects we've chosen. I think this really is the exact vague intuition for which naturality is meant to give an objective correlate. Apr 6, 2016 at 16:52
• @KevinCarlson you mean the intuition I did present is 1- The intuition behind it. 2- and it is vague intuition?
– Ak9
Apr 7, 2016 at 7:40
• @Amir.HKiani yes. That's not to insult it-most intuitions are vague. Apr 7, 2016 at 14:38