Cosilting modules arising from cotilting objects

Let $R$ be a ring. In this paper, we study the characterization of cosilting modules and establish a relation between cosilting modules and cotilting objects in a Grothendieck category. We proved that each cosilting right $R$-module $T$ can be described as a cotilting object in $\sigma [R\mathrm{/}I]$, where $I$ is a right ideal of $R$ determined by $T$ and $\sigma [R\mathrm{/}I]$ is the full subcategory of right $R$-modules, consisting of submodules of $R\mathrm{/}I$-generated modules. Conversely, under some suitable conditions, if $T$ is a cotilting object in $\sigma [R\mathrm{/}I]$, then $T$ is cosilting.