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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 ....

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    It is a natural choice, whenever you define a 2-category in that way, to consider as maps between $\cal C$ and $\cal D$ those functors that induce morphisms of hom-objects; in the case where you enrich over $k$-vector spaces, your functors have to be linear. – 2017-01-04
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    Try to look at [this paper](https://arxiv.org/abs/1409.5723); Domenico and Alessandro give a semi-rigorous definition of the monoidal $(\infty,n)$-category of $n$-vector space, $\text{Vect}_n(k)$; these categories can be linked by a "delooping" operation, in the sense that there are canonical equivalences $\text{End}(1_{\text{Vect}_n(k)}) \cong \text{Vect}_{n-1}(k)$. This should clarify the construction putting it into a more general framework. – 2017-01-04
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    (uhm, Maybe I shouldn't say "semi-rigorous" when speaking of a paper written by my advisor :-) ) – 2017-01-04
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    @FoscoLoregian I took a quick look at the article you linked. I think it's what I need. I suppose you could make your comment with the link the answer to my question. – 2017-01-05
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    I feel that your question still isn't clear. We simply define 2-vector spaces as a 2-category. The Homs between objects of a 2-category form a 1-category, and the 1-categories we choose here are those of $k$-linear functors. Does that get at your question? – 2017-01-05
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    @KevinCarlson Yes that gets at my question. Sorry, I'm trying to learn this stuff on my own so occasionally I need to have obvious ideas verified to make sure I'm on the right track. Furthermore, I've been reading the paper Fosco linked and that has really helped me build my "intuition" for the notion of enriched categories and n-categories. – 2017-01-05

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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.

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    Um? The authors do say right in their introduction that they give "sketches" of definitions. Your advisor would surely be more offended by your describing his work inaccurately than by your agreeing with him about its level of rigor. I know it was just a joke, but students on MSE are not guaranteed to get your jokes. – 2017-01-05
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    As the OP I can say that the solution provided by Fosco completely cleared the issue I was having up. It's very true that I may have not asked the question in the most clear way but I guess that's part of the learning process. I apologize for my lack of clarity, please feel free to mark my question down but mark up Fosco's solution up if you see fit. – 2017-01-05
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    @KevinCarlson, i was only a joke, but I'll edit the answer anyways – 2017-01-05