The notion of conjugation (Kimura 1989 b) is extended to finite-dimensional linear time-varying systems in this paper. In addition, we derive the necessary and sufficient conditions for the existence of stabilizing J-lossless conjugators and J-lossless factorizations of linear time-varying systems. These conditions are shown to exist in the form of a non-negative definite Lyapunov stabilizing solution to a matrix differential Riccati equation. It extends the linear time-invariant results developed by Kimura (1989 b, 1992) to the time-varying case.