A convex cone $C$ is a subsets $C \subseteq V$ of a vector space which is closed under positive linear combinations, i.e. for $\lambda, \mu > 0$ and $u,v \in C$ it is $\lambda u + \mu v \in C$.
An homomorphism between convex cones (in two vector spaces) $\varphi : C_1 \to C_2$ is a map such that
1) it is a homomorphism between the cancelative, commutative semigroups $(C_1, +)$ and $(C_2, \oplus)$ which respects all those laws,
2) it is a homomorphism which respects multiplication by positive scalars and distributivity (i.e. linear for positive linear combinations)
Now are all $n$-dimensional cones isomorphic? For example is the convex cone spannes by two independent vectors isomorphic to $\mathbb R^2$ or $\mathbb R \times \mathbb R^{>0}$? I am not sure because $\mathbb R^2$ is a convex cone but it could not decribed by tupels with just two positive coordinates, but a cone which is induced by two independent vectors could, but I am not sure how to make this precise. But with respect to the above notion of morphism between convex cones maybe they are isomorphic?