Let D be a domain in ℝm+1 and E be a domain in ℝn+1. A pair of a smooth mapping f : D → E and a smooth positive function ϕ on D is called a caloric morphism if ϕ ˙u o f is a solution of the heat equation in D whenever u is a solution of the heat equation in E. We give the characterization of caloric morphisms, and then give the determination of caloric morphisms. In the case of m < n, there are no caloric morphisms. In the case of m = n, caloric morphisms are generated by the dilation, the rotation, the translation and the Appell transformation. In the case of m > n, under some assumption on f, every caloric morphism is obtained by composing a projection with a direct sum of caloric morphisms of ℝn+1.
