Let $FSet$ be the category of finite sets. Let $C$ be a category with binary coproduct and terminal object $*$.
My first question: when does $1\mapsto *$ extend to a functor $FSet\to C$ preserving coproducts? Is this functor an embedding such that $FSet$ is a full subcategory of $C$?
Second question: Denote the above functor by $\iota\colon FSet\to C$. When do we have an isomorphism of categories:
$$ C^{FSet}\cong C/FSet, $$
where $C^{FSet}$ denotes the functor category $FSet\to C$, and $C/FSet$ is the comma/slice category for the functor $\iota$.
Basically, I am asking when a morphism $A\to 2$ gives a decomposition of $A$ (??).
If $\mathcal{C}$ has finite coproducts, then for any object $A$ in $\mathcal{C}$, there is a unique (up unique isomorphism) functor $\mathbf{FinSet} \to \mathcal{C}$ that preserves finite coproducts and sends $1$ to $A$. This is more or less obvious. It is neither full nor faithful in general. (Consider the case where $\mathcal{C}$ is $\mathbf{Set} \times \mathbf{Set}$ and the case where $\mathcal{C}$ is trivial.) However, if $A$ is a connected object, then the embedding is faithful; and if in addition $A$ is the terminal object, then the embedding is fully faithful.
The second question, interpreted generously, has to do with the concept of extensivity. One definition is as follows: an extensive category is a category $\mathcal{C}$ with finite coproducts such that, for all objects $A$ and $B$, the functor $$\mathcal{C}_{/ A} \times \mathcal{C}_{/ B} \to \mathcal{C}_{/ A \amalg B}$$ induced by binary coproduct is an equivalence of categories. Of course, $\mathbf{FinSet}$ is an extensive category. If $\mathcal{C}$ is an extensive category with a terminal object that is connected, then the induced embedding $\mathbf{FinSet} \to \mathcal{C}$ described in the first paragraph preserves finite limits (as well as finite coproducts).