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Here Cartesian category means a category with all finite limits. Let $m:Y\hookrightarrow X$ be a monomorphism, and let ${\epsilon }_X\hookrightarrow X\times PX$ be universal membership for a power object $PX$. Pull it back to $Y\times PX$ and then regard the result as a subobject of $X\times PX$. Its unique classifier is an endomorphism $r:PX\to PX$: on a parameterized subobject it intersects with $Y$. Intersecting twice has the same result as intersecting once, so uniqueness of classification gives $r^2=r$.

Take the equalizer $q:Q\hookrightarrow PX$ of $r$ and the identity, and restrict membership to $Y\times Q$. For any subobject $S\hookrightarrow Y\times Z$, its composite with $m\times 1_Z$ is a subobject of $X\times Z$, hence is uniquely classified by $s:Z\to PX$. Since it is contained in $Y\times Z$, its intersection with that subobject changes nothing, so $rs=s$ and $s$ factors uniquely through $Q$. Conversely every map $Z\to Q$ gives the original kind of family by pulling back the restricted membership. These inverse operations givenaturally in $Z$. Thus $Q$ is a power object of a subobject, even if no power objects other than $PX$ were assumed.

Now consider the slice category $\mathcal{E}/A$, which has finite limits formed by ambient pullbacks and has terminal object $1_A:A\to A$. The constant family $A\times X\to A$ has slice power object $A\times PX\to A$: for $Z\to A$,and a classifier $Z\to PX$ is equivalently the map $Z\to A\times PX$ over $A$. Every family $p:X\to A$ is a subobject of this constant family through its monic graph $(p,1_X):X\hookrightarrow A\times X$. Apply the first construction inside the slice. Consequently every slice of a Cartesian category with power objects again has power objects.

If a Cartesian category has power objects, its slice categories do too. In $\mathcal{E}/A$, the constant family $A\times X\to A$ has power object $A\times PX\to A$, since its product with a family $Y\to A$ is $X\times Y$. An arbitrary family $p:X\to A$ is a subobject of the constant family through its graph $(p,1_X)$, so the power object of a subobject construction applies in the slice.