Here come a few general questions about **Puzzle 283**, I'm trying to see how the abstract concepts and their concrete realizations are linked. They may be highly imprecise though.

1. For vector spaces we have also what is known as [direct sum](https://en.wikipedia.org/wiki/Direct_sum) or direct product, or \\(\oplus\\). Basically it is a Cartesian product of two vector spaces \\(A\\) and \\(B\\), and consists of ordered pairs \\((a, b)\\). Intuitively, it looks like a limit, or at least has its flavor - there are projections from \\(A\times B\\) to spaces \\(A\\) and \\(B\\). Is there something to it, why are we not using it as "tensoring"?

2. It seems that in the case of \\(\oplus\\), \\(I\\) is a 0-dimensional vector space, is this the case?

3. Also we have tensor product \\(\otimes\\) over vector spaces. Speaking abstractly/intuitively about "tensoring", it looks like a colimit, or at least has its flavor - we include or embed two objects into one, or glue them together. Translating this to vector spaces, we see that we indeed don't lose any information, but the resulting space is somewhat "mangled" in the sense that we need to multiply coordinates of the source vector spaces. And to reconstruct the original spaces, we need to solve some equations. What is the motivation of defining "tensoring" as tensor product for **FinVect**?

1. For vector spaces we have also what is known as [direct sum](https://en.wikipedia.org/wiki/Direct_sum) or direct product, or \\(\oplus\\). Basically it is a Cartesian product of two vector spaces \\(A\\) and \\(B\\), and consists of ordered pairs \\((a, b)\\). Intuitively, it looks like a limit, or at least has its flavor - there are projections from \\(A\times B\\) to spaces \\(A\\) and \\(B\\). Is there something to it, why are we not using it as "tensoring"?

2. It seems that in the case of \\(\oplus\\), \\(I\\) is a 0-dimensional vector space, is this the case?

3. Also we have tensor product \\(\otimes\\) over vector spaces. Speaking abstractly/intuitively about "tensoring", it looks like a colimit, or at least has its flavor - we include or embed two objects into one, or glue them together. Translating this to vector spaces, we see that we indeed don't lose any information, but the resulting space is somewhat "mangled" in the sense that we need to multiply coordinates of the source vector spaces. And to reconstruct the original spaces, we need to solve some equations. What is the motivation of defining "tensoring" as tensor product for **FinVect**?