(Co)products and exponentials for Lawvere metric spaces A Lawvere metric space can be thought of as a $\textbf{Cost}$-enriched category where $\textbf{Cost}=([0,\infty],\geq,0,+)$ is a symmetric monoidal preorder (see Chapter 2 of Fong and Spivak for example). I have several questions about $\textbf{Cost}$-enriched categories, and would appreciate help with any of them:

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*How do I define the product of objects in a $\textbf{Cost}$-enriched category ?

*How do I define the coproduct of objects in a $\textbf{Cost}$-enriched category ?

*How do I define the exponential of objects in a $\textbf{Cost}$-enriched category ?

*Are there classes of $\textbf{Cost}$-enriched categories which always have the above structure ? (like categories of functors into $\textbf{Set}$ are always bicartesian closed in the $\textbf{Set}$-enriched case).

 A: I can think of three characterizations of products in a $Cost$-enriched category $X$ (the case of coproducts is dual):

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*adjointness: $X$ has binary products if the functor $\Delta_X : X \to X\otimes X$ has a right adjoint
$$\_\odot\_ : X\otimes X \to X.$$
The object $X\otimes X$ is the product metric space, which has the correct universal property to make $Cost$-categories a monoidal category. Note that this is slightly more than what you asked for, because this way you define when all products exist, not just for a single pair $(x,y)$.


*universal property: given $x,y\in X$, there exists a point $p=x\odot y$ with the following property: let $\alpha_x, \alpha_y$ be respectively the distances $d(p,x), d(p,y)\in [0,\infty]$; then, for every other $z\in X$, the distance $d(z,p)$ is terminal among all real numbers such that
$$ \begin{cases} h + \alpha_x \ge d(z,x) \\ h + \alpha_y \ge d(z,y) \end{cases} $$
all in all this means that $d(z,p)=\max\{d(z,x)-d(p,x),d(z,y)-d(p,y)\}$.
(I think it's just a matter of unwinding the universal property of an adjoint to see that $p = x\odot y$).


*representability: observe that a generic category admits the binary product of two objects $x,y$ when the functor $a\mapsto X(a,x)\times X(a,y)$ is representable, i.e. when (taking into account that $X$ is enriched over $[0,\infty]^\text{op}$)
$$X(a,x\odot y) = X(a,x)\lor X(a,y)$$
i.e. when (taking into account that the hom-object $X(u,v)$ is the real number $d(u,v)$) for every $a\in X$ one has $d(a,x\odot y) = \max\{d(a,x), d(a,y)\}$.
Certainly the three definitions of $x\odot y$ are equivalent; I think I understand the first only in terms of the third. Also, the third generalises easily (well, not so easily if you want your metric to be finite...) to the case of $\kappa$-ary products: given a family $\{x_i\mid i\in\kappa\}$ of elements of $X$, $\bigodot_i x_i$ is a point of $X$ such that for every $a\in X$,
$$\textstyle d(a,\bigodot_ix) = \sup_i d(a,x_i)$$
