  A Sidorenko bigraph is one whose density in a bigraphon $W$ is minimized
precisely when $W$ is constant. Several techniques of the literature to prove
the Sidorenko property consist of decomposing (typically in a tree
decomposition) the bigraph into smaller building blocks with stronger
properties. One prominent such technique is that of $N$-decompositions of
Conlon--Lee, which uses weakly H\"{o}lder (or weakly norming) bigraphs as
building blocks. In turn, to obtain weakly H\"{o}lder bigraphs, it is typical
to use the chain of implications reflection bigraph $\implies$ cut-percolating
bigraph $\implies$ weakly H\"{o}lder bigraph. In an earlier result by the
author with Razborov, we provided a generalization of $N$-decompositions,
called reflective tree decompositions, that uses much weaker building blocks,
called induced-Sidorenko bigraphs, to also obtain Sidorenko bigraphs.
  In this paper, we show that "left-sided" versions of the concepts of
reflection bigraph and cut-percolating bigraph yield a similar chain of
implications: left-reflection bigraph $\implies$ left-cut-percolating bigraph
$\implies$ induced-Sidorenko bigraph. We also show that under mild hypotheses,
the "left-sided" analogue of the weakly H\"{o}lder property (which is also
obtained via a similar chain of implications) can be used to improve bounds on
another result of Conlon--Lee that roughly says that bigraphs with enough
vertices on the right side of each realized degree have the Sidorenko property.
