[efr-KL7V]

If a category has coproducts, then \mathsf {Set} acts on it via S \cdot X = \coprod _{s \in S}X. (This is the \mathsf {Set}-enriched case of what in enriched categories is called a copower or tensor, the dual of the power objects from Example [efr-K3YI]). The morphisms of \mathsf {Para}_\mathsf {Set}(\mathcal {C}) are pairs (I \in \mathsf {Set}, (f_i: X \to Y \in \mathcal {C})_{i \in I}), which compose in the obvious way.

Note that this definition clearly makes sense even if \mathcal {C} does not actually have coproducts. This is an example of another construction which has been called \mathsf {Para}, which takes a monoidal category \mathcal {V} and a \mathcal {V}-enriched category \mathcal {C} and constructs a double category where the morphisms are pairs (J \in \mathcal {V}, J \to \mathcal {C}(X,Y)). It is not hard to see that this also extends to a double category in the same way, but we do not presently know the correct definition of internal enriched object that would replace pseudomonoid actions to replicate our general theory for this case. (note that the literature contains a notion of internal enriched category, Reference [internal-enriched], but these are categories enriched over internal monoidal categories---that is, the ambient category \mathcal {E} is a 1-category and the categorical structure of the base of enrichment \mathcal {V} is formulated on top of this, not as part of the structure of the objects of the category \mathcal {E})