  We begin by explaining how any context-free grammar encodes a functor of
operads from a freely generated operad into a certain "operad of spliced
words". This motivates a more general notion of CFG over any category $C$,
defined as a finite species $S$ equipped with a color denoting the start symbol
and a functor of operads $p : Free[S] \to W[C]$ into the operad of spliced
arrows in $C$. We show that many standard properties of CFGs can be formulated
within this framework, and that usual closure properties of CF languages
generalize to CF languages of arrows. We also discuss a dual fibrational
perspective on the functor $p$ via the notion of "displayed" operad,
corresponding to a lax functor of operads $W[C] \to Span(Set)$.
  We then turn to the Chomsky-Sch\"utzenberger Representation Theorem. We
describe how a non-deterministic finite state automaton can be seen as a
category $Q$ equipped with a pair of objects denoting initial and accepting
states and a functor of categories $Q \to C$ satisfying the unique lifting of
factorizations property and the finite fiber property. Then, we explain how to
extend this notion of automaton to functors of operads, which generalize tree
automata, allowing us to lift an automaton over a category to an automaton over
its operad of spliced arrows. We show that every CFG over a category can be
pulled back along a ND finite state automaton over the same category, and hence
that CF languages are closed under intersection with regular languages. The
last important ingredient is the identification of a left adjoint $C[-] :
Operad \to Cat$ to the operad of spliced arrows functor, building the "contour
category" of an operad. Using this, we generalize the C-S representation
theorem, proving that any context-free language of arrows over a category $C$
is the functorial image of the intersection of a $C$-chromatic tree contour
language and a regular language.
