1)Why embedding of ( not necessarily finite-dimensional) vector spaces $V\rightarrow W$ produces embedding of tensor algebras $T(V)\rightarrow T(W)$? . I can prove it using Hamel basis in W $W$ but is there a nicer ( more functorial ) argument? 2) How to prove the same statement for modules over an algebra instead of vector spaces?

1)Why embedding of ( not necessarily finite-dimensional) vector spaces $V\rightarrow W W$ produces embedding of tensor algebras $T(V)\rightarrow T(W). T(W)$? I can prove it using Hamel basis in W but is there a nicer ( more functorial ) argument? 2) How to prove the same statement for modules over an algebra instead of vector spaces?

1)Why embedding of ( not necessarily finite-dimensional) vector spaces V\rightarrow W produces embedding of tensor algebras T(V)\rightarrow T(W). I can prove it using Hamel basis in W but is there a nicer ( more functorial ) argument? 2) How to prove the same statement for modules over an algebra instead of vector spaces?