TITLE: Intuition for dinatural and extranatural transformations
QUESTION [12 upvotes]: 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.

REPLY [9 votes]: 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.