Abstract. Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C 1 , the second term of the coradical filtration of C. If C = C 1 , we associate with C a K-linear functor H C : C-Comod → H C -Comod that restricts to a representation equivalence, where H C is a coradical square complete hereditary bipartite K-coalgebra such that every simple H Ccomodule is injective or projective. Here H C -comod• sp is the full subcategory of H C -comod whose objects are finite-dimensional H C -comodules with projective socle having no injective summands of the form(see Theorem 5.11). Hence, we conclude that a coalgebra C with C = C 1 is left pure semisimple if and only if H C is left pure semisimple. In Section 6 we get a diagrammatic characterisation of coradical square complete coalgebras C that are left pure semisimple. Tameness and wildness of such coalgebras C is also discussed.1. Introduction. Throughout this paper we fix an arbitrary field K and we use the coalgebra representation theory notation and terminology introduced in [14] [39] for the coalgebra and comodule terminology. In particular, given a finite-dimensional K-algebra R, we denote by mod(R) the category of all finite-dimensional R-modules.Let C be a K-coalgebra with comultiplication ∆ and counit ε. We recall that a left C-comodule is a K-vector space X together with a K-linear map δ X : X → C ⊗ X such that (∆ ⊗ id X )δ X = (id C ⊗ δ X )δ X and (ε ⊗ id X )δ X is the canonical isomorphism X ∼ = K ⊗ X, where ⊗ = ⊗ K . Given a left C-comodule X, we denote by X 0 = soc X the socle of X, that is, the sum of all simple C-subcomodules of X.