Category of Sets and Bag-valued functions I asked here about the Category of sets and set-valued functions, and it turns out it to be equal to REL (Category of sets and Relations),so a good studding point to study that category. 
Now, It happens to me that I need to also take a look at the Category of Sets and Bag-Valued functions (i.e., Nodes are sets, and arrows are multi-valued functions.) I should add that I am interested in finite sets.  
Question:
Can anybody please provide a hint how can I start studding such category (that is, the Category of Sets and Bag-Valued functions). Is there any specific literature out there that already studied this category, in terms of its limits, and collimates, and etc.
 A: Since you want to focus on finite sets, I will assume that a bag can only contain finitely many copies of each element. Let $\mathbb{N}_{> 0}$ be set of positive integers. Then a "bag-valued function" from $X$ to $Y$ is an ordinary map $X \to \mathbb{N}_{> 0} \times Y$. In order to define a category, we must give a composition law, and again, it suffices to observe that a monad structure on $\mathbb{N}_{> 0} \times (-) : \mathbf{Set} \to \mathbf{Set}$ would give such a composition law via the Kleisli category construction.
The easiest way to give a monad structure on an endofunctor of the form $M \times (-)$ for some set $M$ is to give a monoid structure on $M$. The resulting Kleisli composition law is easy to describe: given $f : X \to M \times Y$ and $g : Y \to M \times Z$, the Kleisli composite $g * f : X \to M \times Z$ is defined as follows: if $f (x) = (y, m)$ and $g (y) = (z, n)$, then $(g * f) (x) = (z, m n)$.
There is an obvious monoid structure on $\mathbb{N}_{> 0}$, namely multiplication, so this gives the desired category of sets and "bag-valued functions". It is equivalent to the full subcategory of the category of sets with an $\mathbb{N}_{> 0}$-action spanned by the sets with a free $\mathbb{N}_{> 0}$-action. As such, this category has coproducts. It also has pullbacks, but it does not have a terminal object. I doubt there is much more that can be said.
