carrier is the image under an $F$-coalgebra morphism of an $F$-coalgebra with free and finitely generated carrier.

Moreover, note that zig-zags which start (or end) with incoming arrows instead of outgoing ones, can also be allowed since a zig-zag of this form can be turned into one of the form by appending identity maps.

Some concrete monads and functors {#some-concrete-monads-and-functors .unnumbered}
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We deal with the following base categories.

-   The category ${\text{$\mathbb S$-{\text{\tt{SMOD}}}}}$ of semimodules over a semiring $\mathbb S$ induced by the monad $T_{\mathbb S}$ of finitely supported maps into $\mathbb S$, see, e.g., [@manes.mulry:2007 Example 4.2.5].

-   The category  of positively convex algebras induced by the monad of finitely supported subprobability distributions, see, e.g., [@doberkat:2006; @doberkat:2008] and [@pumpluen:1984].

For $n\in\mathbb N$, the free algebra with $n$ generators in ${\text{$\mathbb S$-{\text{\tt{SMOD}}}}}$ is the direct product $\mathbb S^n$, and in  it is the $n$-simplex $\Delta^n=\{(\xi_1,\ldots,\xi_n)\mid \xi_j\geq 0,\sum_{j=1}^n\xi_j\leq 1\}$.

Concerning semimodule-categories, we mainly deal with the semirings $\mathbb N$, $\mathbb Q_+$, and $\mathbb R_+$, and their ring completions $\mathbb Z$, $\mathbb Q$, and $\mathbb R$. For these semirings the categories of $\mathbb S$-semimodules are

-   ${{\text{\tt{CMON}}}}$ of commutative monoids for $\mathbb N$,

-   ${{\text{\tt{AB}}}}$ of abelian groups for $\mathbb Z$,

