I have one issue with the Jacquet Langlands correspondence. The Weyl law for $H$ modulo a congruence subgroup and the Weyl law for cocompact groups are different. So why does this not contradict this functoriality? What am I missing?
I have not yet studied the Jacquet Langlands correspondence explicitly yet. How explicit are the lifts, about the level etc.? I know that there is not an expansion formula for cocompact groups available as we have it for groups with a parabolic element.
Update: After a reading a little bit, I found a paper which focuses exactly on the first part of the question and also gives references for the second part of the question:
Risager, Morten S. Asymptotic densities of Maass newforms. J. Number Theory 109 (2004), no. 1, 96–119.