<p>We give an explicit formula to express the cohomological pullback functors of Hodge modules under closed immersions of smooth varieties using Verdier specializations and <i>V</i>-filtrations of Kashiwara and Malgrange. This was locally obtained by the first two authors assuming the existence of global defining functions. We also give a quite simplified proof of the theorem reducing to the monodromical case via the Verdier specialization and using induction on codimension.</p>