special relativity - Transferring from vectors and tensors in $(-+++)$ signature to $(+---)$ signature?

Minkowski space with the signature $(+---)$ can be described by $\Bbb{C}^{1,3}$ whilst with the signature $(-+++)$ by $\Bbb{C}^{3,1}$ (I am using $\Bbb{C}$ instead of $\Bbb{R}$ to allow for Wick rotations). Given a standard vector (or tensor) in one of these spaces: $$\vec v=v_i\hat e^i, \quad T=T^{ij} \hat e_i \otimes \hat e_j$$ what is the standard isomorphism(?) used to form the equivalent vector/tensor in the other?

My guess is that for vectors the isomorphism is defined to keep the $v^i$ components the same in both spaces - is this correct?.