Let \(A\) be a covariant \(k\)-tensor field and \(F\colon M\to N\) smooth. Then the pullback in local coordinates is given by
\begin{equation*} F^*(A_{i_1,\ldots,i_k}dx^{i_1}\otimes \cdots \otimes dx^{i_k}) = (A_{i_1,\ldots,i_k} \circ F)d(x^{i_1}\circ F)\otimes \cdots \otimes d(x^{i_k}\circ F), \end{equation*}where \((x^i)\) are smooth coordinates.
That means, if we want to compute the pullback for different coordinates, we just apply them and calculate. For instance see Example 12.29 in [@lee2013smooth_manifolds].