I'm reviewing free modules and tensor products as part of a foray back into abstract algebra. I notice that the Wikipedia article on the tensor product has undergone some changes in the past few years, leading to some confusion on my part.
I seem to recall once seeing a free vector space $F(V)$ over some initial vector space $V$ over a field $K$ defined as $$ F(V) = \left\lbrace f \in K^V : \text{$f$ is linear and has finite support} \right\rbrace $$ the set of $K$-linear maps from $V$ to $K$ with finite support, with the set of indicator functions as a basis. This is a bit stronger than the usual definition of $F(S)$ for $S$ an arbitrary set in that linearity is imposed. $$ F(S) = \left\lbrace f \in K^S : \text{$f$ has finite support} \right\rbrace $$
Of course, if we define $F(S)$ and $F(V)$ in category-theoretic terms of Hom-sets, then the two definitions are equivalent, since the $K$-linear maps from $V$ to $K$ are just the morphisms $V \to K$ of $\mathbf{KVec}$, just as the functions from $S$ to $K$ are the morphisms $S \to K$ in $\mathbf{Set}$.
\begin{align} F(V) &= \left\lbrace f \in \mathrm{Hom}_{\mathbf{KVec}}(V,K) : \text{$f$ has finite support} \right\rbrace \\ F(S) &= \left\lbrace f \in \mathrm{Hom}_{\mathbf{Set}}(S,K) : \text{$f$ has finite support} \right\rbrace \end{align}
Are the two (equivalent) definitions of $F(V)$ valid? Is the Hom-set definition of the free functor the most general?
Any resources on free objects/functors would be appreciated, as I'm trying to learn some category theory as well.