  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 $\sigma[R/I]$, 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 $R/I$-generated
modules. Conversely, under some suitable conditions, if $T$ is a cotilting
object in $\sigma[R/I]$, then $T$ is cosilting.
