In general, for any monoidal category $(\mathcal M, \otimes, I)$ and a monoid $(m, *, i) \in \mathcal M$, the slice category $\mathcal M/m$ inherits the monoidal structure pointwise, i.e. the tensor product is given by taking morphisms $f : x \to m$ and $g : y \to m$ to the morphism $x \otimes y \xrightarrow{f \otimes g} m \otimes m \xrightarrow{*} m$, and the unit is given by $i : I \to m$.

In your example, we take $(\mathbf{Set}, \times, 1)$ to be the category of sets equipped with cartesian monoidal structure and $(m, *, i)$ to be the power set monoid $(\mathcal P A, \cup, \varnothing)$.

Abstractly, we can see resulting monoidal structure on $\mathbf{Set}/\mathcal{P}A$ to be given by the following composite

$$(\mathbf{Set}/\mathcal{P}A)^2 \simeq (\mathbf{Set}^{\mathcal{P}A})^2 \xrightarrow{\langle-,-\rangle} (\mathbf{Set}^2)^{({\mathcal{P}A}^2)} \xrightarrow{\times^{({\mathcal{P}A}^2)}} \mathbf{Set}^{({\mathcal{P}A}^2)} \simeq \mathbf{Set}/({\mathcal{P}A}^2) \xrightarrow{\mathbf{Set}/\cup} \mathbf{Set}/{\mathcal{P}A}$$

where the equivalences are given by the correspondence between indexed sets and $\mathbf{Set}$-functors from discrete categories. (The unit is the same, replacing $2$ by $0$.)

In general, for any monoidal category $(\mathcal M, \otimes, I)$ and a monoid $(m, *, i) \in \mathcal M$, the slice category $\mathcal M/m$ inherits the monoidal structure pointwise, i.e. the tensor product is given by taking morphisms $f : x \to m$ and $g : y \to m$ to the morphism $x \otimes y \xrightarrow{f \otimes g} m \otimes m \xrightarrow{*} m$, and the unit is given by $i : I \to m$.

In your example, we take $(\mathbf{Set}, \times, 1)$ to be the category of sets equipped with cartesian monoidal structure and $(m, *, i)$ to be the power set monoid $(\mathcal P A, \cup, \varnothing)$.

Abstractly, we can see resulting tensor product on $\mathbf{Set}/\mathcal{P}A$ to be given by the following composite

$$(\mathbf{Set}/\mathcal{P}A)^2 \simeq (\mathbf{Set}^{\mathcal{P}A})^2 \xrightarrow{\langle-,-\rangle} (\mathbf{Set}^2)^{({\mathcal{P}A}^2)} \xrightarrow{\times^{({\mathcal{P}A}^2)}} \mathbf{Set}^{({\mathcal{P}A}^2)} \simeq \mathbf{Set}/({\mathcal{P}A}^2) \xrightarrow{\mathbf{Set}/\cup} \mathbf{Set}/{\mathcal{P}A}$$

where the equivalences are given by the correspondence between indexed sets and $\mathbf{Set}$-functors from discrete categories.

The unit is given similarly by

$$1 \xrightarrow{1} \mathbf{Set} \simeq \mathbf{Set}/1 \xrightarrow{\mathbf{Set}/i} \mathbf{Set}/{\mathcal{P}A}$$