Pullback of a covector (${\varphi }^{\ast }\eta =\eta \circ d\varphi$)

The pullback of a covector is the dual linear map to the differential of a smooth map. In coordinates its components are $({\varphi }^{\ast }\eta {)}_i=({\partial }_i{y}^{\alpha }){\eta }_{\alpha }$. The same rectangular Jacobian defines vector pushforward, with its other index contracted. No invertibility is needed.