  Formally verifying the properties of formal systems using a proof assistant
requires justifying numerous minor lemmas about capture-avoiding substitution.
Despite work on category-theoretic accounts of syntax and variable binding,
raw, first-order representations of syntax, the kind considered by many
practitioners and compiler frontends, have received relatively little
attention. Therefore applications miss out on the benefits of category theory,
most notably the promise of reusing formalized infrastructural lemmas between
implementations of different systems. Our Coq framework Tealeaves provides
libraries of reusable infrastructure for a raw, locally nameless representation
and can be extended to other representations in a modular fashion. In this
paper we give a string-diagrammatic account of decorated traversable monads
(DTMs), the key abstraction implemented by Tealeaves. We define DTMs as monoids
of structured endofunctors before proving a representation theorem a la
Kleisli, yielding a recursion combinator for finitary tree-like datatypes.
