Conjugates of rational equivariant holomorphic maps of symmetric domains

Let tau: D --> D' be an equivariant holomorphic map of symmetric domains associated to a homomorphism rho: G --> G' of semisimple algebraic groups defined over Q. If Gamma subset of G(Q) and Gamma' subset of G'(G) are torsion-free arithmetic subgroups with rho(Gamma) subset of Gamma', the map tau induces a morphism phi: Gamma\D --> Gamma'\D' of arithmetic varieties and the rationality of tau is defined by using symmetries on D and D' as well as the commensurability groups of Gamma and Gamma'. An element sigma is an element of Aut(C) determines a conjugate equivariant holomorphic map tau(sigma) :D-sigma --> D-'sigma of tau which induces the conjugate morphism phi(sigma): (Gamma\D)(sigma) - -> (Gamma'\D)(sigma) of phi. We prove that tau(sigma) is rational if tau is rational.