does the h3 have an oil/tranny cooler?

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Cooler is built into the radiator as in all GM transmission, except for those that are equipped with external coolers. If doing a lot of towing, I would recommend an external large volume cooler, but under normal operation, the internal is enough. (NOTE: My recommendation for external is mine, GM specs show it is good enough for towing up to rated capacity. So take the recommendation for what is is worth, a personal opinion.)