  We study tensors on Lie groupoids suitably compatible with the groupoid
structure, called {\em multiplicative}. Our main result gives a complete
description of these objects only in terms of infinitesimal data. Special cases
include the infinitesimal counterparts of multiplicative forms, multivector
fields and holomorphic structures, obtained through a unifying and conceptual
method. We also give a full treatment of multiplicative vector-valued forms,
particularly Nijenhuis operators and related structures.
