  In this article we prove Adams inequality with exact growth condition in the
four dimensional hyperbolic space $\mathbb{H}^4,$
  \begin{align}
  \int_{\mathbb{H}^4} \frac{e^{32 \pi^2 u^2} - 1}{(1 + |u|)^2} \ dv_g \leq C
||u||^2_{L^2({\mathbb{H}^4})}.
  \end{align} for all $u \in C^{\infty}_c(\mathbb{H}^4) $ with
$\int_{\mathbb{H}^4} (P_2 u) u \ dv_g \leq 1.$
  We will also establish an Adachi-Tanaka type inequality in this settings.
Another aspect of this article is the P.L.Lions lemma in the hyperbolic space.
We prove P.L.Lions lemma for the Moser functional and for a few cases of the
Adams functional on the whole hyperbolic space.
