I am reading something where this is used extensively, but it is not defined anywhere and no references are given, and I can't find any.



In a k-linear category, what is the tensor product between a hom space and an object?
If $\mathcal{C}$ is a $k$-linear category, $X \in \mathcal{C}$, and $V$ is any $k$-vector space (in particular, it could be a hom of two objects in $\mathcal{C}$), then $V \otimes X$ (sometimes written $V \odot Y$ to avoid confusion with a monoidal structure) is the object representing the functor $\mathcal{C} \to \operatorname{Vect}$ sending $Y \in \mathcal{C}$ to $\operatorname{Hom}\_{\operatorname{Vect}}(V, \mathcal{C}(X, Y))$, if such an object exists. This is a special case of the notion of [copower](http://ncatlab.org/nlab/show/copower).


In the case where $\mathcal{C}$ is a tensor category with internal homs, the construction agrees with the tensoring by the internal hom.


