  We investigate several interrelated foundational questions pertaining to the
study of motivic dga's of Dan-Cohen--Schlank [8] and Iwanari [13]. In
particular, we note that morphisms of motivic dga's can reasonably be thought
of as a nonabelian analog of motivic cohomology. Just as abelian motivic
cohomology is a homotopy group of a spectrum coming from K-theory, the space of
morphisms of motivic dga's is a certain limit of such spectra; we give an
explicit formula for this limit --- a possible first step towards explicit
computations or dimension bounds. We also consider commutative comonoids in
Chow motives, which we call ``motivic Chow coalgebras''. We discuss the
relationship between motivic Chow coalgebras and motivic dga's of smooth proper
schemes.
  As a small first application of our results, we show that among schemes which
are finite \'etale over a number field, morphisms of associated motivic dga's
are no different than morphisms of schemes. This may be regarded as a small
consequence of a plausible generalization of Kim's relative unipotent section
conjecture, hence as an ounce of evidence for the latter.
