We present a variant of a universality result of Rodl [On universality of graphs with uniformly distributed edges, Discrete Math. 59 (1986), no. 1-2, 125-134] for sparse, $3$-uniform hypergraphs contained in strongly jumbled hypergraphs. One of the ingredients of our proof is a counting lemma for fixed hypergraphs in sparse ``pseudorandom uniform hypergraphs, which is proved in the companion paper [Counting results for sparse pseudorandom hypergraphs I].
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