The question is in the title, here is my motivation:

$\require{AMScd}$Let $(\mathcal C,\otimes,I)$ be a monoidal symmetric closed category. Then, the tensor product commutes with colimits, and if $\mathcal C$ has infinite coproducts, the object $N:=\displaystyle\coprod_{n\in\mathbb N}I$ has the following properties :

- it is a natural numbers object (NNO) in the sense of Lawvere, ie it has natural morphisms $0:I\to N$ and $S:N \to N$ such that any given $I\to X\to X$ uniquely defines a morphism $f:N\to X$ such that the following diagram commutes

$$\begin{CD} I @>0>> N @>S>> N\\ @| @VfVV @VfVV\\ I @>>> X @>>> X \end{CD}$$

- it is also the free monoid on $I$, ie the initial object in the category of monoids under $I$

If such a coproduct does not exist, a natural numbers object (if it exists) is the free monoid on $I$, but the converse is not necessarily true (or at least, I haven't been able to prove it). To better understand what might happen, It would help to see a workable example of such a category.

Of course for such an example to be interesting, I would like to have a category where an NNO or the free monoid exists.

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