The terminology "contravariant" is not obsolete in mathematics, it's used all over, so we all need to know it. We still say hom is contravariant in the first argument and covariant in the second, but what we _mean_ is that it's a functor \\(\text{hom} : \mathcal{C}^{\text{op}} \times \mathcal{C} \to \mathbf{Set} \\), with an op in the first slot and not the second.