Hi there in wiki the tensor algebra is defined w.r.t. the adjoint of the forgetful functor rather than the forgetful functor itself - why so?
Besides, does the existence of such algebras for every vector spaces correlate to the existence of the adjoint functor?
Moreover, can somebody carefully check wether I'm getting everything right in here: 
The universal property is defined via the forgetful functor: $\mathrm{U}:\mathrm{Alg}\to\mathrm{Vec}$
This defines its adjoint functor: $\mathrm{T}:\mathrm{Vec}\to\mathrm{Alg}$