So I just encountered the word functoriality in my script for linear algebra, in the chapter multilinear mappings. I don't really know what it means and after some research it turns out it's a term in category theory and the definition and explanation i found don't seem to be the same as the one in my script:
Given linear map $$f_i:V_i' \rightarrow V_i$$ and $$g:W \rightarrow W'$$ in a K vector space. They induce a linear map:
$$Mult_K(V_1,....,V_r;W) \rightarrow Mult_K(V_1',....,V_r';W), \phi \rightarrow g\circ\phi\circ(f_1 \times...\times f_r)$$
with the following proposition:
Consider the basis $B_i$ of $V_i$ and $C$ of $W$. Consider a system of coefficients $a_{b_1,...,b_r}^c \in K$ for all $b_i \in B_i$ and $c \in C$ with the property $\forall b_i \in B_i: |\{c\in C |a_{b_1,...,b_r}^c \ne 0\}| < \infty$. Then there exisits exactly one mulinear map $\phi: V_1 \times ... \times V_r \rightarrow W$, such that for all $b_i \in B_i$
$$\phi(b_1,...,b_r)=\sum_{c\in C}'a_{b_1,...b_r}^cc$$
Conversely, every multilinear mapping $\phi: V_1 \times ... \times V_r \rightarrow W$ has distinct coefficients $a_{b_1,...b_r}^c$.
So I don't really understand these two propositions. I downloaded a few pdf books on multilinear algebra and tensor products, and none of them has these two propositions. So would appreciate some explanation.
The first one seems to be similar to the universal problem but I got lost with the composition with $(f_1 \times ... \times f_r)$. The second one i don't really understand the notation for the coefficients $a_{b_1,...b_r}^c$, but it just follows from the definition of a multilinear mapping, turning multiple vectors into a number in the corresponding field?
Any help much appreciated!