中文摘要：

我们在 Morita-Takeuchi 上下文 coalgebra 上调查 comodule 表示范畴并且学习 Gorensteinness。而且，我们明确地在 Morita-Takeuchi 上下文 coalgebra 上决定所有 Gorenstein injective comodules 并且在 Gorenstein coalgebras 讨论本地化。特别地，我们描述它的加布里埃尔发抖并且当 Morita-Takeuchi 上下文 coalgebra 是基本的时，执行一些例子。

英文摘要：

We investigate the comodule representation category over the Morita-Takeuchi context coalgebra F and study the Gorensteinness of F. Moreover, we determine explicitly all Gorenstein injective comodules over the Morita-Takeuchi context coalgebra F 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.