## Mnemonic for how left and right duals interact with Homs

Suppose you have a rigid monoidal category. This means you have a left dual and a right dual. By Frobenius reciprocity, you can move objects from one side of a Hom to the other side at the price of dualizing (e.g. something like $\mathrm{Hom}(X \otimes Y, Z) = \mathrm{Hom}(X, Z \otimes Y^*$). However this comes in four flavors: you can tensor on the left or on the right and you can move from the input to output or vice-versa. For each of these flavors you need to remember whether to take left dual or right dual. Half of these are easy to remember: if you know how to move from input to output, then you have to use the other kind of dual to move back. But I can never remember consistent conventions for everything.

So does anyone have some easy-to-remember clear mnemonic for dealing with this?

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I find that thinking in string diagram pictures is easiest for me. The identification of homs comes from taking a map $X \otimes Y \to Z$ and bending one of the strings around to the other side, as in the picture below.

What we get is a map $X \to Z \otimes Y^{\ast}$. How do you know that this is $Y^{\ast}$ and not ${^{\ast}}Y$? Well, I call $Y^{\ast}$ the left right dual (maybe other people call it the right left dual), and it's the one where the arrow on the string goes from right to left, at least the way I draw the diagrams. The other way to remember it is that the ${\ast}$ goes on the inside in the evaluation pairing (and hence on the outside in the coevaluation).

I don't think people will ever agree on conventions for which way string diagrams go, or which one is the left dual and which one is the right dual, but I can at least be internally consistent with these conventions.

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Of course! That's very nice. (Aside: the reason that no one can agree about the direction to read diagrams all goes back to the confusion about the order of function composition. If you want "left to right" to agree with "top to bottom" but use the usual "right to left" convention for function composition, then you have to read "bottom to top.") – Noah Snyder Aug 3 2011 at 17:39
I'm typically a progressive on these issues; new notation should correct the mistakes of the past, not entrench them. Besides, when you see a composite of several functions written out, don't your eyes immediately jump to the innermost function anyway? I'm pretty sure mine do. When I want to understand what a function does, I follow the input to the output. Hence, having the input on the top makes much more sense to me. – Evan Jenkins Aug 3 2011 at 19:06
@Evan Jenkins, surely your eyes immediately jumping to the innermost function is an artifact of mathematical training, not necessarily of what's natural? Thus this, too, should be subject to correction instead of entrenchment! (I'm mostly playing devil's advocate, but I do have a soft spot for Herstein's use of function application on the right.) – L Spice Aug 3 2011 at 19:10
And now that you mention it, if I'm going to insist on top-to-bottom string diagrams, I should probably also insist on calling $V^{\ast}$ the right dual. The naming of left vs. right dual should be compatible with the naming of left vs. right adjoint, but whether the left adjoint appears on the left in the unit or in the counit depends on whether we compose functions left-to-right or right-to-left! So the more progressive choice would be to call $V^{\ast}$ the right dual, although this makes the arrow mnemonic the reverse of my intuition. (It fixes the $\ast$ reversal, though, so it's a wash.) – Evan Jenkins Aug 3 2011 at 19:10
Except that the mouth doesn't point the right direction (unlike say the arms that do the acting in a semidirect product). – Noah Snyder Aug 3 2011 at 23:45