Abstract
For a category 𝒞, we investigate the problem of when the coproduct ⊕ and the product functor ∏ from 𝒞 I to 𝒞 are isomorphic for a fixed set I, or, equivalently, when the two functors are Frobenius functors. We show that for an Ab category 𝒞 this happens if and only if the set I is finite (and even in a much general case, if there is a morphism in 𝒞 that is invertible with respect to addition). However, we show that ⊕ and ∏ are always isomorphic on a suitable subcategory of 𝒞 I which is isomorphic to 𝒞 I but is not a full subcategory. If 𝒞 is only a preadditive category, then we give an example that shows that the two functors can be isomorphic for infinite sets I. For the module category case, we provide a different proof to display an interesting connection to the notion of Frobenius corings.
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