  This paper is a coalgebra version of arXiv:1703.04266 and a sequel to
arXiv:1607.03066. We present the definition of a pseudo-dualizing complex of
bicomodules over a pair of coassociative coalgebras $\mathcal C$ and $\mathcal
D$. For any such complex $\mathcal L^\bullet$, we construct a triangulated
category endowed with a pair of (possibly degenerate) t-structures of the
derived type, whose hearts are the abelian categories of left $\mathcal
C$-comodules and left $\mathcal D$-contramodules. A weak version of
pseudo-derived categories arising out of (co)resolving subcategories in
abelian/exact categories with enough homotopy adjusted complexes is also
considered. Quasi-finiteness conditions for coalgebras, comodules, and
contramodules are discussed as a preliminary material.
