2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/148793The general problem for consistency between arbitrary transports along paths in fibre bundles and bundle morphisms between them is formulated and investigated. The special case of one fibre bundle, its morphism and transport along paths acting in it is considered. The consistency between linear transports along paths in a vector bundle and a Hermitian structure in it is studied.21 standard (11pt, A4) LaTeX 2.09 pages. For other papers of this series view http://www.inrne.bas.bg/mathmod/bozhome/ . New author's e-mail address: bozho@inrne.bas.bgDifferential Geometry53B99, 53C99, 55R99, 53B15text