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. LaTeX 2e with tikz-cd, 42 pages, 10 commutative diagrams; v.2: details and comments added in the proof of Lemma 2.6(b), two references added; v.3: references updated; v.4: the reference to arXiv:2101.10797 added; v.5: several misprints corrected; v.6: many misprints corrected, the numbering of sections shifted to agree with the journal version