Transfer of complex structures and topological invariance of property of being holomorphic

The following question by V. I. Arnold is answered in affirmative. Let X, Y, and Z be three complex manifolds of equal dimension, let p: X → Y be a universal covering, and let g: Y → Z be a nondegenerate holornorphic mapping. Assume that the term Y in the chain X\xrightarrowpY\xrightarrowgZ X\xrightarrow{p}Y\xrightarrow{g}Z is “forgotten,” while the complex structures on X and Z are changed so that the mapping g ∘ p remains holomorphic. Can one recover the “forgotten” term Y? Bibliography: 2 titles.