  We show that the regular patterns of Getzler (2009) form a 2-category
biequivalent to the 2-category of substitudes of Day and Street (2003), and
that the Feynman categories of Kaufmann and Ward (2013) form a 2-category
biequivalent to the 2-category of coloured operads (with invertible 2-cells).
These biequivalences induce equivalences between the corresponding categories
of algebras. There are three main ingredients in establishing these
biequivalences. The first is a strictification theorem (exploiting Power's
General Coherence Result) which allows to reduce to the case where the
structure maps are identity-on-objects functors and strict monoidal. Second, we
subsume the Getzler and Kaufmann--Ward hereditary axioms into the notion of
Guitart exactness, a general condition ensuring compatibility between certain
left Kan extensions and a given monad, in this case the
free-symmetric-monoidal-category monad. Finally we set up a biadjunction
between substitudes and what we call pinned symmetric monoidal categories, from
which the results follow as a consequence of the fact that the hereditary map
is precisely the counit of this biadjunction.
