I'm reading the On the Classification of Topological Field Theories and have a question about the use of enriching categories in the definition of a strict 2-category found on page 9. These are rather simple questions but I want to make sure I understand before reading further.
Is it correct to say that the author is using enriched categories in an effort to make functors into objects while giving them structure?
In the case of $\mathcal{C}, \mathcal{D}\in\operatorname{Vect_2}(k)$ is it true that $\operatorname{Maps_{Vect_2(k)}}(\mathcal{C}, \mathcal{D})$ is a category of $k$-linear functors precisely because the $k$-linear structure was inherited during the process of enrichment using $\operatorname{Vect}(k)$?
EDIT: I added the little bit in bold for clarity ....
Try to look at this paper; Domenico and Alessandro give a sketch of definition of the monoidal $(∞,n)$-category of $n$-vector space, $Vect_n(k)$; these categories can be linked by a "delooping" operation, in the sense that there are canonical equivalences $End(1_{Vect_n(k)})≅Vect_{n−1}(k)$. This should clarify the construction putting it into a more general framework.
:-) there's a little difference between the comment and the answer.