Re: Finite sets and injective maps

I thank everyone who answered. The answer is "the free symmetric monoidal category with an initial unit generated by one object". Some were curious to know what I am doing with the category. In a study of generalizations of relational databases with a student of mine we found that a good category to use for describing schemata (shapes of relations) is the category whose objects look like freely generated coproducts and the morphisms are injective functions. So I now know that this category is the freely generated symmetric monoidal category with an initial unit generated by the set of types that may appear in a schema. Thank you. Best regards, Andrej