For example, a tensor field $T=T_{a,b}^{\ \ \ \ c}$ on a manifold $M$ should be thought as $ T_{a,b}^{\ \ \ \ c}dx^{a}\otimes dx^{b}\otimes \frac{\partial}{\partial x^{c}} $ with local coordinate $x=(x^{1},\dots,x^{n})$ of $M$. The transformation rule of tensors is nothing but the change of local coordinates of $M$, i.e. if one changes the local coordinate $x\mapsto y$ with Jacobian matrix $g$, then the tensor above transforms like $ T_{a,b}^{\ \ \ \ c} \mapsto T_{A,B}^{\ \ \ \ C}=T_{a,b}^{\ \ \ \ c}g_{A}^{\ \ a}g^{\ \ b}_{B}g^{\ \ C}_{c}. $ How should one think of the transformation rule of spinor fields? What local change reflects the transformation rule?
My question is a bit ambiguous, but I hope everyone can understand what I try to say. Thank you.