# 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).
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 '11 at 19:10