Halve the head of cabbage, and cut off the bottom part of the core. Then cut each half crosswise into ½-inch ribbons.
Heat the coconut oil over medium high heat until melted, and add onion. Cook onion for 2 minutes, stirring occasionally, and add cabbage. Toss to coat with oil. Season lightly with salt and liberally with pepper, and toss again. Let sit for 4-5 minutes, until the underside of the cabbage starts to color. Use a stiff spatula to flip the cabbage, and let sit for another few minutes, until it begins to color, then repeat the process.
When cabbage is well-colored and the thickest parts of the leaves are tender, after 15-20 minutes, add the soy sauce to the pan, turn off the heat, and scrape up anything stuck to the bottom of the pan. Squeeze lime over the top, and toss one last time. Season with additional salt and pepper, if desired.