  We show the equivalence of the definitions of very strict $CD(K,N)$
-condition defined, on one hand, using (only) the entropy functionals, and on
the other, the full displacement convexity class $\mathcal{DC}_N$. In
particular, we show that assuming the convexity inequalities for the critical
exponent implies it for all the greater exponents. We also establish the
existence of optimal transport maps in very strict $CD(K,N)$ -spaces with
finite $N$.
