I've seen several times on the Internet the pretty cool expression that \begin{equation}\label{1}\mathrm df=\sum_{i,j}{\partial f^i\over \partial x^j}{\partial \over \partial y^i}\otimes\mathrm dx^j\tag{1}\end{equation} here $f:M\to N$ is a differentiable map between manifolds, $x^j$ and $y^i$ a set of coordinates on $M$ and $N,$ respectively.
To me, it seems that equation \eqref{1} identifies the differential(i.e. the tangent mapping or the pushforward) $\mathrm df$ as a $(1,1)$ tensor field on $M$. But there is some ambiguity, since the "pointwise"(and the accurate) version of equation \eqref{1} should be \begin{equation}\label{2}\mathrm df(p)=\sum_{i,j}{\partial f^i\over \partial x^j}(p){\partial \over \partial y^i}\Big|_{\color{red}{f(p)}}\otimes\mathrm dx^j\Big|_p\tag{2}\end{equation} So the RHS of \eqref{1} is not a section of $TM\otimes T^*M.$
Question: I wonder how to construct a proper bundle such that the RHS of \eqref{1} could be understood as a section of such bundle. In particular, I'm looking for a reference book that describes in detail the identification of the differential $\mathrm df$ with a tensor field. Any help would be appreciated!