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.



Example(s) of monoidal symmetric closed category with NNO without infinite coproducts?
The effective topos is an example of a (locally!) cartesian closed category with NNO where infinite coproducts do *not* exist. Indeed, the main feature of the NNO $N$ in the effective topos is that the endomorphisms of $N$ are precisely the computable functions. As such, $N$ cannot be (isomorphic to) $\coprod\_{n \in \mathbb{N}} 1$: if that were the case, then there would be uncountably many endomorphisms of $N$. (Of course, $\coprod\_{n \in \mathbb{N}} 1$ does not even exist in the effective topos; as you say, if it existed, it would have to be the NNO.)


Here is a closely related construction. Suppose we have a countable model $M$ of Zermelo set theory, or even just Mac Lane set theory. (For instance, by Skolem, we may take $M$ to be a countable elementary substructure of $V\_{\omega + \omega}$.) Then the category of sets in $M$ is an elementary topos (so locally cartesian closed) with NNO, but its NNO cannot be $\coprod\_{n \in \mathbb{N}} 1$, for the same reason.


