  We provide a short characterization of $p$-asymptotic uniform smoothability
and asymptotic uniform flatenability of operators and of Banach spaces. We use
these characterizations to show that many asymptotic uniform smoothness
properties pass to injective tensor products of operators and of Banach spaces.
In particular, we prove that the injective tensor product of two asymptotically
uniformly smooth Banach spaces is asymptotically uniformly smooth. We prove
that for $1<p<\infty$, the class of $p$-asymptotically uniformly smoothable
operators can be endowed with an ideal norm making this class a Banach ideal.
We also prove that the class of asymptotically uniformly flattenable operators
can be endowed with an ideal norm making this class a Banach ideal.
