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We investigate the comodule representation category over the Morita-Takeuchi context coalgebra Gamma and study the Gorensteinness of Gamma. Moreover, we determine explicitly all Gorenstein injective comodules over the Morita-Takeuchi context coalgebra Gamma and discuss the localization in Gorenstein coalgebras. In particular, we describe its Gabriel quiver and carry out some examples when the Morita-Takeuchi context coalgebra is basic.
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