Fuck universal properties, what do you mean that my algebraic structure is a functor HOW DOES THAT EVEN WORK

(trying to understand how given a vector space, we can construct a tensor algebra, so "T is a functor from K-Vect to K-Alg")

Like is the act of being able to turn a vector space into T(V) what makes it into a functor?

(if you respond, please do explain this in a way that someone who only knows what categories and vaguely what functors are can understand)

-Ryan
Oh also, relevant wikipedia

-Ryan