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There might be some point to reconsidering the so-called tensor algebra, insofar as it is really a construction of the "universal k-algebra" AMattached to a module M over a (e.g.) commutative ring k. That is, it is the image under an adjoint functor, $\Hom_{k-mod}(V,FB)\isom Hom_{k-mod}(V,FB) \Hom_{k-alg}(AV,B)$, cong Hom_{k-alg}(AV,B)$, for k-algebras , where F is the forgetful functor from k-algebras to k-modules. Thus, the colimit of the tensor modules is a construction, proving existence. There is the usual categorical virtue that the properties do not depend on the construction. 1 [made Community Wiki] There might be some point to reconsidering the so-called tensor algebra, insofar as it is really a construction of the "universal k-algebra" AMattached to a module M over a (e.g.) commutative ring k. That is, it is the image under an adjoint functor,$\Hom_{k-mod}(V,FB)\isom \Hom_{k-alg}(AV,B)\$, for k-algebras , where F is the forgetful functor from k-algebras to k-modules.