Heat soy milk and coconut milk over medium heat for 5–7 minutes until warm but not boiling, stirring gently to prevent scorching and ensure even heating.
Mix cornstarch with a small amount of warm milk until smooth, then add it with agar-agar to the saucepan and stir for 2–3 minutes until slightly thickened.
Allow the mixture to cool for 15–20 minutes until lukewarm, ensuring it is not hot to preserve the live cultures.
Whisk in the vegan yogurt starter thoroughly so the cultures are evenly distributed throughout the mixture.
Transfer to jars and ferment in a warm place for 8–12 hours until set and tangy.
Refrigerate for 2–3 hours before serving to allow the yogurt to fully thicken and develop flavor.