= Power object of a subobject
{title2=$PY=\operatorname{Eq}(r,1_{PX})$}
For a <monomorphism> $Y\hookrightarrow X$ in a <Cartesian category>, intersect the universal membership relation on $X\times PX$ with $Y\times PX$. The resulting <subobject> is classified by $r:PX\to PX$, and $r^2=r$. The <equalizer> of $r$ and the identity represents precisely families of subobjects of $X$ contained in $Y$, hence is a <power object> for $Y$. This construction only requires the power object of $X$ and finite limits.
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