  We consider the holomorphic unramified mapping of two arbitrary finite
bordered Riemann surfaces. Extending the map to the doubles $X_1$ and $X_2$ of
Riemann surfaces we define the vector bundle on the second double as a direct
image of the vector bundle on first double. %% We choose line bundles of
half-order differentials $\Delta_1$ and $\Delta_2$ so that the vector bundle
$V^{X_2}_{\chi_\ad} \otimes \Delta_2$ on $X_2$ would be the direct image of the
vector bundle $V^{X_1}_{\chi_\ao} \otimes \Delta_1$. We then show that the
Hardy spaces $H_{2, J_1(p)} (S_1,V_{\chi_\ao} \otimes \Delta_1)$ and
$H_{2,J_2(p)} (S_2,V_{\chi_\ad} \otimes \Delta_2)$ are isometrically
isomorphic. Proving that we construct an explicit isometric isomorphism and a
matrix representation $\chi_\ad$ of the fundamental group $\piod(X_2, p_0)$
given a matrix representation $\chi_\ao$ of the fundamental group $\piod(X_1,
p'_0)$. %% On the basis of the results of \cite{vin} and Theorem
\ref{theorem_1} proven in the present work we then conjecture that there exists
a covariant functor from the category ${\cal RH}$ of finite bordered surfaces
with vector bundle and signature matrices to the category of Kre\u{\i}n spaces
and isomorphisms which are ramified covering of Riemann surfaces.
