We address the problem of making Conformal Prediction (CP) intervals locally adaptive. Most existing methods focus on approximating the object-conditional validity of the intervals by partitioning or re-weighting the calibration set. Our strategy is new and conceptually different. Instead of re-weighting the calibration data, we redefine the conformity measure through a trainable change of variables, $A \to φ_X(A)$, that depends explicitly on the object attributes, $X$. Under certain conditions and if $φ_X$ is monotonic in $A$ for any $X$, the transformations produce prediction intervals that are guaranteed to be marginally valid and have $X$-dependent sizes. We describe how to parameterize and train $φ_X$ to maximize the interval efficiency. Contrary to other CP-aware training methods, the objective function is smooth and can be minimized through standard gradient methods without approximations.