Consider the full subcategory of $\mathbf{Set}$ consisting of the singleton $1$ and countable sets. (Originally this came from the powers $\mathbb{N}^{\times k}$ and the morphisms between them interpreted as a Lawvere theory.) What are the symmetric monoidal functors from this subcategory to $\mathbf{Set}$?

Consider the full subcategory of $\mathbf{Set}$ consisting of the singleton $1$ and countable infinite sets. (Originally this came from the powers $\mathbb{N}^{\times k}$ and the morphisms between them interpreted as a Lawvere theory.) What are the symmetric monoidal functors from this subcategory to $\mathbf{Set}$?

Consider the full subcategory of $\textbf{Set}$ consisting of the singleton $1$ and countable sets. (Originally this came from the powers $\mathbb{N}^{\times k}$ and the morphisms between them interpreted as a Lawvere theory.) What are the symmetric monoidal functors from this subcategory to $\textbf{Set}$?

Consider the full subcategory of $\mathbf{Set}$ consisting of the singleton $1$ and countable sets. (Originally this came from the powers $\mathbb{N}^{\times k}$ and the morphisms between them interpreted as a Lawvere theory.) What are the symmetric monoidal functors from this subcategory to $\mathbf{Set}$?