In some famous papers like Gaitsgory's and Beilinson-Bernstein's , the authors used the nearby cycle functor $\Psi$ to schemes over an affine line (sometime even over an algebraic curve), and they always referred to Exposé XIII of SGA 7. However in loc. cit., I just find the definition of nearby cycle functor for schemes over strict traits (spectrum of strict henselian local rings). My question is what is the precise definition of nearby cycle functor over an affine line? Is there any authentic reference for this type of nearby cycle?

In some famous papers like Gaitsgory's "Construction of central elements in the affine Hecke algebra via nearby cycles" and Beilinson-Bernstein's "A proof of Jantzen conjectures", the authors used the nearby cycle functor $\Psi$ to schemes over an affine line (sometime even over an algebraic curve), and they always referred to Exposé XIII of SGA 7. However in loc. cit., I just find the definition of nearby cycle functor for schemes over strict traits (spectrum of strict henselian local rings). My question is what is the precise definition of nearby cycle functor over an affine line? Is there any authentic reference for this type of nearby cycle?