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](http://www.math.jhu.edu/~eriehl/yoneda.pdf), 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. 



Morphisms of cotensors
It's pretty simple, really: using the universal property of $A^U$, one may define maps $E^i: E^V \to E^U$ and $p^V: E^V \to B^V$, and similarly maps $B^i: B^V \to B^U$ and $p^U: E^U \to B^U$. 


For example, if $I$ is the monoidal unit of $\mathcal{V}$, one has a canonical arrow $\theta\_{E, V}: I \to \mathcal{V}(V, \mathcal{C}(E^V, E))$ mated to the name of the identity $I \to \mathcal{C}(E^V, E^V)$, and then one may form the composite 


$$I \stackrel{\theta\_{E, V}}{\to} \mathcal{V}(V, \mathcal{C}(E^V, E)) \stackrel{\mathcal{V}(i, 1)}{\to} \mathcal{V}(U, \mathcal{C}(E^V, E)) \cong \mathcal{C}(E^V, E^U)$$ 


which names the map $E^i: E^V \to E^U$. The other arrows mentioned above may be constructed by similar considerations. 


It is not hard to see by the yoga of enriched universal mapping properties that the cotensor thus becomes a $\mathcal{V}$-functor $\mathcal{V}^{op} \otimes \mathcal{C} \to \mathcal{C}$ in two variables, meaning in particular that there is a commutative square 


$$\begin{array}{ccc} 
E^V & \stackrel{p^V}{\to} & B^V \\ 
E^i \downarrow & & \downarrow B^i \\ 
E^U & \underset{p^U}{\to} & B^U
\end{array}$$


and so there is an induced map $E^V \to E^U \times\_{B^U} B^V$ which is this thing you are asking about. 


