A note on exhaustion of hyperbolic complex manifolds

The purpose of this article is to investigate a hyperbolic complex manifold $M$ exhausted by a pseudoconvex domain $\mathrm{\Omega }$ in ${\mathbb{C}}^n$ via an exhausting sequence $\{f_j\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to M\}$ such that $f_j^{-1}(a)$ converges to a boundary point ${\xi }_0\in \mathrm{\partial }\mathrm{\Omega }$ for some point $a\in M$.