Let $\mathcal{C}$ be a category with coproducts and zero morphisms. Then we have projections $\bigoplus_{i \in I} M_i \to M_i$. For every object $T$ they induce a map

$\hom(T,\bigoplus_{i \in I} M_i) \to \prod_{i \in I} \hom(T,M_i).$

Now we can make the condition on $\mathcal{C}$ that this map is always injective. Notice that if $\mathcal{C}$ has products, the condition means that $\bigoplus_{i \in I} M_i \to \prod_{i \in I} M_i$ is a monomorphism.

Of course, this is satisfied in many familiar categories (but it fails for the category of groups). I would like to know what is known about this condition, in particular if it already has a name and which familiar properties of $\mathcal{C}$ imply it.

For example, it holds when $\mathcal{C}$ is linear, $\mathcal{C}$ has products and monomorphisms are closed under directed colimits. In particular this is true when $\mathcal{C}$ is linear and locally finitely presentable.


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