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Conceptually, natural transformations make perfect sense. What is the intuition behind dinatural and extranatural transformations?

Added: I'm looking for conceptual intuition, not something alone the lines of "bending a natural transformation". For instance, my conceptual intuition for natural transformations is that naturality in $A$ of an arrow $FA\rightarrow GA$ means the arrow does not (in a sense) depend on the object $A$. The way to see this is the fact naturality squares commute independently of the specific element of $\mathsf{Hom}(A,B)$, which, by a Yoneda-type argument, exactly means we don't care about the object $A$ itself.

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    $\begingroup$ Have you looked at this? There are a few helpful pictures. The first section of the article on dinatural transformations could be helpful too. $\endgroup$ – Najib Idrissi Mar 12 '15 at 10:33
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    $\begingroup$ @NajibIdrissi I edited my question to be more specific. The nlab offers nice graphical intuition and motivation, but I don't have any conceptual feel for extranaturality, e.g how to detect it. That is what I'm looking for. $\endgroup$ – Arrow Mar 12 '15 at 11:45
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    $\begingroup$ I do not believe there is any intuition to be had. Extranaturality is merely a technical condition that is the next best thing to naturality in certain contexts. $\endgroup$ – Zhen Lin Mar 12 '15 at 11:55
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Let $ F,G : {\bf C} \to {\bf D} $ be functors. A natural transformation is just a set of maps $$ \{ \alpha_C : F(C) \to G(C) \}_{C \in {\bf C}} $$ which satisfy a coherence condition: for each morphism $f : C \to C' $ in $ {\bf C} $, the following diagram commutes: $$ \require{AMScd} \begin{CD} F(C) @>{\alpha}>> G(C)\\ @V{F(f)}VV @V{F(f)}VV \\ F(C') @>{\alpha}>> G(C') \end{CD}. $$ Natural transformations are not the be all and end all of coherence in category theory. For example, the pentagon axiom is a coherence condition which appears when defining monoidal categories. Extranatural transformations are just sets of maps which satisfy a coherence condition that looks a little bit like the one for natural transformations.

Indeed, let $ T : {\bf C}^{\rm op} \times {\bf C} \to {\bf D} $ be a functor and $D$ an object in ${\bf D}$. An extranatural transformation $D \to T $ is a set of maps $$ \{ \beta_A : D \to T(C,C) \}_{C \in {\bf C}} $$ such that for each map $ f : B \to C$ the following diagram commutes: $$ \require{AMScd} \begin{CD} D @>{\beta}>> T(B,B)\\ @V{\beta}VV @V{T(1,f)}VV \\ T(C,C) @>{T(f,1)}>> T(B,C) \end{CD}. $$ This type of extranatural transformation is important because the end of $T$ is the terminal extranatural transformation $D \to T$.

Once you are happy with this case, the general definition is not much worse. This is where the "string diagrams" come in. These string diagrams are different from the ones used to notate 2-categories.

Take a bipartite graph where each vertex has valence 1. Label the vertices like in the following diagram:

extranatural domain

Choose functors $ T : {\bf C}^{\rm op} \times {\bf A} \times {\bf C} \times {\bf D} \to {\bf Q} $ and $U : {\bf D} \times {\bf E} \times {\bf A} \times {\bf E}^{\rm op} \to {\bf Q} $. Then an extranatural transformation $ T \to U$ consists of maps $$ \{ \gamma_{A,C,D,E} : T(C,A,C,D) \to U(D,E,A,E) \} $$ which satisfy a bunch of coherence conditions of the two types above. You can vertically compose two extranatural transformations exactly when the domain bipartite graphs glue together. This was proved by Eilenberg and Kelly in the paper "A generalization of the functorial calculus".

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Dinaturality and extranaturality are just the correct kinds of naturality for some constructions. Here is perhaps the simplest example: if $V$ is a vector space and $V^{\ast} = \text{Hom}(V, k)$ is its dual space, then I think you will agree that there is an evaluation pairing

$$\text{ev} : V \otimes V^{\ast} \to k$$

and that it is in some sense natural in $V$. But in what sense? The problem with just using naturality is that $V$ is a covariant functor of $V$ but $V^{\ast}$ is a contravariant functor. Nevertheless if you just try to write down with your bare hands exactly in what sense the evaluation pairing is natural, you'll end up writing down a special case of extranaturality.

$\text{ev}$ also has a natural interpretation in terms of string diagrams which is described e.g. in these blog posts.

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