= Solution
Use the PDF's covector-first convention: its space $\mathcal T_l^k$ consists of <tensor fields> in
$$
(T^*M)^{\otimes k}\otimes(TM)^{\otimes l}.
$$
This reverses the order in which some texts list <tensor type>. For a finite-dimensional real <vector space> $V$, define the <tensor contraction> first on decomposable tensors:
$$
C_j^i(\alpha_1\otimes\cdots\otimes\alpha_k\otimes v_1\otimes\cdots\otimes v_l)
=\alpha_i(v_j)\,\alpha_1\otimes\cdots\widehat{\alpha_i}\cdots\otimes\alpha_k
\otimes v_1\otimes\cdots\widehat{v_j}\cdots\otimes v_l.
$$
The hats mean omission, with the other factors left in their original order. The formula is a <multilinear map> of its individual factors, so the <universal property of a tensor product> gives a unique <linear map> with this formula.
Apply it with $V=T_pM$ at every $p$. Evaluation of a <covector> on a vector is basis independent: under a change of frame, one factor transforms by a <matrix> and the other by its inverse transpose, and the <matrices> cancel in the pairing. Thus the fiber maps agree on overlapping <vector bundle trivializations>. In a local <dual basis>, the coefficients of the contracted tensor are finite sums of coefficients with the selected covariant and contravariant indices set equal. They remain smooth. Hence
$$
\boxed{C_j^i:\mathcal T_l^k\longrightarrow\mathcal T_{l-1}^{k-1}}
$$
is a globally defined smooth contraction for $k,l\geq1$. It is also linear over $C^\infty(M)$.
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