If you want to know more how functors are used in Я I recommend you to check out What makes a Functor feel like Hom? article.
The notion of Hom functors in Я is used for bifunctors, whose parameters behave in the same manner as it was a Hom-functor (left - contravariantly, right - covariantly).
I recommend you to spend some time starring at this Natural Transformation between Hom functors. Isn’t it beautiful?