Let $\mathcal{V}$ be a closed monoidal category, and $\mathscr{C}$ be a category enriched over $\mathcal{V}$. One says that the power or cotensor of an objec $A \in \mathscr{C}$ by an object $U \in \mathcal{V}$ is an object $A^U$ of $\mathscr{C}$ with a natural isomorphism $$\mathscr{C}(B, A^U) \cong \mathcal{V}(U, \mathscr{C}(B, A)),$$ where $B \in \mathscr{C}$, and $\mathcal{V}(-, -)$ is the inner hom of $\mathcal{V}$.

Riehl and Verity, in their third of a series of papers towards an axiomatic theory of $(\infty, 1)$-categories, define an $\infty$-cosmos to be some sort of category, the details of which do not really matter to my question. My question does, however, regard one certain part of their definition. They require that the class of isofibrations be closed under forming the Leibniz cotensor of two maps, and I am having a bit of difficulty seeing how this map is defined. Specifically, the Leibniz cotensor of a map $p: E \to B$ in $\mathscr{C}$ by a map $i: U \to V$ in $\mathcal{V}$ is a map

$$ i\, \widehat{\pitchfork}\, p: E^V \to E^U \times_{B^U} B^V. $$

Certainly, this map would follow by the universal property of pullbacks given maps $E^V \to E^U$ and $E^V \to B^V$ that satisfy the requisite commutativity condition. My question is as follows:

How does one define these maps? Indeed, a good definition of these maps is necessary even to form the pullback.

Any insight would be greatly appreciated.