  The purpose of this article is twofold. The first aim is to characterize an
$n$-dimensional hyperbolic complex manifold $M$ exhausted by a sequence
$\{\Omega_j\}$ of domains in $\mathbb C^n$ via an exhausting sequence
$\{f_j\colon \Omega_j\to M\}$ such that $f_j^{-1}(a)$ converges to a boundary
point $\xi_0 \in \partial \Omega$ for some point $a\in M$. Then, our second aim
is to show that any spherically extreme boundary point must be strongly
pseudoconvex.
