right adjoint by forgetting relational symbols

Dear all, I need some references for this problem. Suppose we have a locally presentable category C axiomatized by a limits theory T, so C=Mod(T). Let us forget some relational symbols in T and all axioms containing these relational symbols. One obtains a theory T'. There is a forgetful functor Mod(T) --> Mod(T'). Does this functor have always a right adjoint ? If not, what conditions must we add ? Thanks in advance. pg.