  Generalizing work of Doi and of Idrissi, we define a coHochschild homology
theory for chain coalgebras over any commutative ring and prove its naturality
with respect to morphisms of chain coalgebras up to strong homotopy. As a
consequence we obtain that if the comultiplication of a chain coalgebra $C$ is
itself a morphism of chain coalgebras up to strong homotopy, then the
coHochschild complex $\cohoch (C)$ admits a natural comultiplicative structure.
In particular, if $K$ is a reduced simplicial set and $C_{*}K$ is its
normalized chain complex, then $\cohoch (C_{*}K)$ is naturally a
homotopy-coassociative chain coalgebra. We provide a simple, explicit formula
for the comultiplication on $\cohoch (C_{*}K)$ when $K$ is a simplicial
suspension.
  The coHochschild complex construction is topologically relevant. Given two
simplicial maps $g,h:K\to L$, where $K$ and $L$ are reduced, the homology of
the coHochschild complex of $C_{*}L$ with coefficients in $C_{*}K$ is
isomorphic to the homology of the homotopy coincidence space of the geometric
realizations of $g$ and $h$, and this isomorphism respects comultiplicative
structure. In particular, there a isomorphism, respecting comultiplicative
structure, from the homology of $\cohoch(C_{*}K)$ to $H_{*}\op L|K|$, the
homology of the free loops on the geometric realization of $K$.
