  We show that the construction due to Leinster and Weber of a generalized
Lawvere theory for a familially representable monad on a (co)presheaf category,
and the associated ``nerve'' functor from monad algebras to (co)presheaves,
have an elegant categorical description in the double category
$\mathbb{C}\mathbf{at}^{\#}$ of categories, cofunctors, familial functors, and
transformations. In $\mathbb{C}\mathbf{at}^{\#}$, which also arises from
comonoids in the category of polynomial functors, both a familial monad and a
(co)presheaf it acts on can be modeled as horizontal morphisms; from this
perspective, the theory category associated to the monad is built using left
Kan extension in the category of endomorphisms, and the nerve functor is
modeled by a single composition of horizontal morphisms in
$\mathbb{C}\mathbf{at}^{\#}$. For the free category monad $path$ on graphs,
this provides a new construction of the simplex category as $\Delta :=
\lens{path}{path \circ path}$. We also explore the free Eilenberg-Moore
completion of $\mathbb{C}\mathbf{at}^{\#}$, in which constructions such as the
free symmetric monoidal category monad on $\mathbf{Cat}$ can modeled using the
rich language of polynomial functors.
