Ag.algebraische Geometrie – A naive question about Serres GAGA

To let $ X, Y $ to be a complex, affine strain and $ X ^ h, Y ^ h $ the analysis schemes assigned to them. To let $ f: X ^ h to Y ^ h $ Be a biholomorphic analytical map of analytic schemes. Is it true then (via Serre's GAGA) that there is a 1-1 match between the connected sheaves $ X $ and that on $ Y $? In other words, the withdrawal of a coherent sheaf is on $ Y ^ h $, again a coherent sheaf over $ X ^ h $? Every note is welcome.