While I understand the concept of product category, I do wonder if there is something like a coproduct category, where we have functors
$$ P: \mathcal{C} \to \mathcal{C} \coprod \mathcal{D} \\ Q: \mathcal{C} \to \mathcal{C} \coprod \mathcal{D} \\ $$
fulfilling a universal property:
$$ \forall R: \mathcal{C} \to \mathcal{O}, T: \mathcal{D} \to \mathcal{O}. \exists! F: \mathcal{C} \coprod \mathcal{D} \to \mathcal{O}. PF = R \wedge QF = T. $$
In fact this should also happen to category of diagrams for the coproduct as a limit. As I understand it, there are certain categories, where the product category and the product expressed in limits doesn't match, which expresses the need for the product category.
Is there such a need for a coproduct category?
Yes, there exists a coproduct category $\mathcal C\sqcup\mathcal D$ of categories $\mathcal C$ and $\mathcal D$, namely it is their disjoint union, with the natural embeddings, just like in the category of sets, graphs or topological spaces.
The product category $\mathcal C\times\mathcal D\ $ is one that satisfies the universal property of products in the category $\mathcal Cat$ of small categories.