Canonical morphism from coproduct to product - Questions 3-6: Infinite index sets In a pointed category $C$, for any family $\{C_i\}_{i \in I}$ of objects, such that both their product and their coproduct exist, there is a canonical morphism
$$
\varphi: \coprod\limits_{i \in I} C_i \to \prod\limits_{i \in I} C_i
$$
defined by the components
$$
\text{pr}_j \circ \varphi \circ \text{ins}_i = \delta_{ji}
$$
for all $i,j \in I$, where $\delta_{ji}$ is the identity if $i = j$ and the zero morphism otherwise. If this morphism is an isomorphism for finite $I$, the (co) product is called a biproduct, but apart from this important case I find very little information about it in the literature. 
By many examples, I see the following for infinite $I$:


*

*$\varphi$ can be monic (e.g. $\text{Set}_*, \text{Mod}_R$) or not ($\text{Group}$).

*$\varphi$ is commonly not epic (the only counterexample I know of is the (co)product of zero objects).


These these yield my questions:


*

*Is there a case where $\varphi$ is epic with nontrivial (co)factors? If there is:

*Are there cases where (for nontrivial cofactors) $\varphi$ is epic and


*

*not monic.

*monic but not isic.

*even isic.



Can we make some triviality assertions on $C$, if one of these hold?
 A: *

*Yes, take the opposite of a pointed category where you know it's monic, e.g. $\text{Ab}^{op}$. By Pontryagin duality this is equivalent to the category of compact Hausdorff abelian groups. The product is the cartesian product but the coproduct is compactification of the direct sum. 

*As above, in $\text{Ab}^{op}$ it's epic but not monic. I'm going to separate out your other two subquestions as full questions.

*For an example which is both monic and epic but not an iso I think we can again consider the category $\text{Ban}_1$ of Banach spaces and weak contractions. It's worth spelling out in some detail what the product and coproduct are here. For a family $X_i$ of Banach spaces, the coproduct is the completion of the direct sum under the "$\ell^1$ norm" $\| (x_i) \|_1 = \sum_i \| x_i \|$. The product is the subspace of the cartesian product for which the "$\ell^{\infty}$ norm" $\| (x_i) \|_{\infty} = \sup_i \| x_i \|$ is finite. I think the map $\coprod_i X_i \to \prod_i X_i$ is injective (but I haven't checked this) with dense image (note that the norm on the codomain is not the induced norm on the image), which means it's both monic and epic.

*For an example which is even an iso I think we can consider the category of suplattices. I believe, but have not checked, that coproducts and products of arbitrary arity agree here. This is at least consistent with the nLab's claim that the free suplattice on a set $X$ (equivalently, the coproduct of $|X|$ copies of the free suplattice on a singleton) is the powerset $2^X$, which is also the product of $|X|$ copies of the free suplattice on a singleton. 
