I want to define a category of multisets, $Multi$. To do this, I take the ambient category $SET$, and represent the multisets as functions. So, the objects $Multi$ are functions. We define composition via spans in $SET$. If I have two multisests $f: C \rightarrow A$ and $g: C \rightarrow B$, then a morphism from $A$ to $B$ is the span $\langle f,g \rangle$. Next, I need to compose the spans, so I use pullbacks of two spans. Is this a valid construction?
It has been pointed out that I am defining a rather limited category that might not be interesting. Instead, I might want to define the morphisms as commutative squares. So, if $f,g$ are two objects, then a morphism is a pair of functions, $i,j$, where $i:C \rightarrow C$, $j: A \rightarrow B$ and $ g \cdot i = j \cdot f$.