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.
http://math.uchicago.edu/~ejenkins/misc/hom-tensor.png
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 dual (maybe other people call it the right 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.