categories: Structure Preserving: Definition?

[<math@antiquark.com>]

Hello,
I've sent the following message to sci.math, but haven't
received a clear answer. I've also tried sci.math.research,
but the moderator bounced the posting. Possibly someone 
here can help?

Derek.
===============================================

I'm working through the following paper, trying to learn a bit
more about category theory:

Matrices, Monads and the Fast Fourier Transform
http://citeseer.nj.nec.com/jay93matrice.html

I this paper, the author explains vectors in categorical
notation:

"Vectors are distinguished from lists because their length
is given as part of their structure, represented by a morphism
(function) #: VA -> N."

What this means is that the morphism '#' will produce the
length of vector.

However, does this violate one of the requirements that a
morphism must preserve the structure of an object?  A vector
is a sequence of elements, and an integer is only a single
value. Does this mean that an integer has the same structure
as a vector?

Or does "structure preserving morphism" mean something
different?

Thanks,

Derek.