If a bidual space $X^{**}$ is isomorphic to a dual space $Y^*$, is there any relation between $(X, \rho|_X)^*$ and $Y$?

Question

If a bidual space $X^{**}$ is isomorphically renormed to a dual space $Y^*$, in other words there exists a norm $\rho$ s.t.$(X^{**},\rho )=Y^*$, then is there any relation between $(X, \rho|_X)^*$ and $Y$?

Thank you in advance.