I have come across two definitions of a smooth tensor field of type $(r,s)$ and I was wondering if someone could explain how they are equivalent.
Definition 1: A smooth tensor field on a differentiable manifold $M$ of type $(r,s)$ is a map $A: C_r^\infty(TM)\rightarrow C_s^\infty(TM)$ which is multilinear over the commutative ring $C^\infty(M)$. Here, $C^\infty(TM)$ is the set of smooth vector fields on $M$, $C_r^\infty(TM)$ is the r-fold tensor product of $C^\infty(TM)$ over $C^\infty(M)$ and $C_0^\infty=C^\infty(M)$.
Definition 2: A smooth tensor field of type $(r,s)$ on a smooth manifold $M$ is a $C^\infty(M)$-multilinear map $A: C^\infty(T^*M)\times ... \times C^\infty(T^*M) \times C^\infty(TM)\times ... \times C^\infty(TM) \rightarrow C^\infty (M)$, so the product of $r$ covector fields and $s$ vector fields are being sent to $C^\infty(M)$.