The formal dual to the pushout product (frequently considered in the context of enriched model category theory) does not have a widely established name, but plausibly deserves to be called the pullback powering operation. Note that it sometimes called pullback hom.

More precisely, a pushout product is defined with respect to a functor of the form $E_1\times E_2 \to E_3$, while a pullback power is defined with respect to a functor of the form $E_2^{op} \times E_3\to E_1$ or $E_1^{op}\times E_3 \to E_2$, of the sort that would be the right adjoints in a two-variable adjunction.