Verdugo is the "right hand" of Ramon Salazar. Verdugo is seen many times in cutscenes, but isn't actually fought until after Ramon captures Ashley Graham. Before the fight begins, Leon makes the joke to Ramon, "Your right hand comes off?".
The best way to deal with Verdugo is to shoot tanks of liquid nitrogen while he is standing by them. This will leave him open either for some gunshots, or a good kick. He is very weak while frozen and will eventually be shattered. It should be noted that a single rocket launcher blast will shatter him when frozen.
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