In a category $\mathcal{C}$, a *generalized element* of an object $A$ means a morphism to $A$. It follows from Yoneda lemma that the object $A$ is determined by the collection of generalized of elements of $A$.

In a monoidal category $(\mathcal{C},\otimes)$, it seems more natural to consider generalized elements of a tensor product $A\otimes B$ in the form $a\otimes b$, meaning a tensor product of two morphisms, one to $A$, another to $B$. So, my first questions are

**Is the tensor product $A\otimes B$ determined by generalized elements of the form $a\otimes b$?****Let $f,g\colon A\otimes B\to C$ be two morphisms. If for all generalized elements of the form $a\otimes b$, the compositions $f(a\otimes b)=g(a\otimes b)$, is it true that $f=g$?**

In nLab, it mentioned that generalized elements of the form $I\to A$, where $I$ is the tensor unit, is important in the theory of enriched category. However, the functor $A\mapsto\mathrm{Hom}(I,A)$ may not be full or faithful. Hence those generalized elements cannot determine the objects or morphisms in general. So

**Why generalized elements of the form $I\to A$ important in enriched category theory regardless its application to define underlying categories?**