  We fill a gap in the literature regarding `transport of structure' for
(n+2)-angulated, n-exact, n-abelian and n-exangulated categories appearing in
(classical and higher) homological algebra. As an application of our main
results, we show that a skeleton of one of these kinds of categories inherits
the same structure in a canonical way, up to equivalence. In particular, it
follows that a skeleton of a weak (n+2)-angulated category is in fact what we
call a strong (n+2)-angulated category. When n=1 this clarifies a technical
concern with the definition of a cluster category. We also introduce the notion
of an n-exangulated functor between n-exangulated categories. This recovers the
definition of an (n+2)-angulated functor when the categories concerned are
(n+2)-angulated, and the higher analogue of an exact functor when the
categories concerned are n-exact.
