More on abundance of cosilting modules

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