  Let EK be the simplicial suspension of a pointed simplicial set K. We
construct a chain model of the James map, $\alpha_{K} : CK \to \Omega CEK$. We
compute the cobar diagonal on $\Omega CEK$, not assuming that $EK$ is
1-reduced, and show that $\alpha_{K}$ is comultiplicative. As a result, the
natural isomorphism of chain algebras $TCK \cong \Omega CK$ preserves
diagonals.
  In an appendix, we show that the Milgram map, $\Omega (A \otimes B) \to
\Omega A \otimes \Omega B$, where A and B are coaugmented coalgebras, forms
part of a strong deformation retract of chain complexes. Therefore, it is a
chain equivalence even when A and B are not 1-connected.
