In https://www.sciencedirect.com/science/article/pii/0001870891900378 section 6 a cotilting module T over an algebra A is said to be strong in case $\hat{add(T)}$ coincides with the subcategory of modules having finite injective dimension.

Here $\hat{add(T)}$ is just the full subcategory of all module $M$ such that there is an exact sequence $0 \rightarrow T_n \rightarrow ... \rightarrow T_0 \rightarrow M \rightarrow 0$ with $T_i \in add(T)$.

Let $\Gamma=End_A(T)$. Proposition 6.5. in this article states that T is a strong cotilting module iff every simple $\Gamma$-module is contained in T as a $\Gamma$-module.

Question: What means "every simple $\Gamma$-module is contained in T as a $\Gamma$-module"?

Im a bit confused here. For example let $A$ be an algebra of finite positive global dimension and $T=A$ (then $\Gamma=A$) with dominant dimension at least one (for example the quiver algebra of the quiver of Dynkin type $A_2$).

Then $T$ is a strong cotilting module but soc(T)=soc(A) does not contain every simple module as a right or left module.

So I think that I have a thinking error or I use the wrong definition of "contained".