(put this is the section on bases)

when looking at the ring operations, which equalities are
"on the nose",
and which are "up to isomorphism"?
(that is, which commute with our usual identifications)

addition is:
 - associative on the nose
 - commutative requires isomorphism
   (obviously)

tensor is:
 - ass on nose
 - comm requires iso

distributive is:
 - right distributive on the nose
   that is, $(V \oplus V') \otimes W
     = (V \otimes W) \oplus (V' \otimes W)$
 - left distributive up to iso!
   but $V \otimes (W \oplus W')
     \cong (V \otimes W) \oplus (V \otimes W')$
...a good way to see this is:
$(\bR^n \oplus \bR^m) \otimes_\bR \bC$
vs
$\bC \otimes_\bR (\bR^n \oplus \bR^m)$
in the former, the bases concatenate nicely, but in the latter you
need to shuffle.
This is easy to understand: you put a copy of the right side
into each slot on the left -- so if you add on the left, it's fine,
but if you add on the right, you need to shuffle so the slots line up
(this is poorly written -- illustrate)

Incidentally, this is why $e_1,ie_1,\dots,e_n,ie_n$
is often a better basis for the underlying real vector space
of a complex vector space than
$e_1,\dots,e_n,ie_1,\dots,ie_n$,
though the latter results in cuter matrices, and is useful for some
purposes.

$\Hom(V,W) = V^* \otimes W$ is on the nose

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Note that the tensor product is usually defined by a universal
property (that's the categorical way), which only defines it up to
(admittedly unique) isomorphism;
showing it actually exists requires a construction.
Making the associativity strict (IE, on the nose) requires a
careful construction (or abstract argument).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

It is claimed in
Schauenburg, "Turning Monoidal Categories into Strict Ones"

that it's possible (for abstract reasons) to make the tensor product
of modules associative on the nose.
This is presumably useful for some abstract reasoning, and excuses our
casualness

However, he doesn't give a concrete description, so it's unclear
whether this has a nice concrete realization, or if it's just pushing
bumps out of a carpet (IE, making hugely many arbitrary choices, and
all the other realizations of tensor products follow).

