Intuition for dinatural and extranatural transformations

Solution 1:

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.

Solution 2:

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:

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".