  We describe a construction that to each algebraically specified notion of
higher-dimensional category associates a notion of homomorphism which preserves
the categorical structure only up to weakly invertible higher cells. The
construction is such that these homomorphisms admit a strictly associative and
unital composition. We give two applications of this construction. The first is
to tricategories; and here we do not obtain the trihomomorphisms defined by
Gordon, Power and Street, but only something equivalent in a suitable sense.
The second is to Batanin's weak omega-categories.
