Interaction With Coproducts. Let $f\colon A\to A'$ and $g\colon B\to B'$ be maps of sets. We have
\[ \webleft (f\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}g\webright )^{-1}\webleft (U'\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}V'\webright )=f^{-1}\webleft (U'\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}g^{-1}\webleft (V'\webright ) \]
for each $U'\in \mathcal{P}\webleft (A'\webright )$ and each $V'\in \mathcal{P}\webleft (B'\webright )$.