  For a category C we investigate the problem of when the coproduct $\bigoplus$
and the product functor $\prod$ 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. Moreover,
this is true even in a much more general case, if there is a morphism in C that
is invertible with respect to the addition of morphisms. If C does not have
this property we give an example to see that the two functors can be isomorphic
for infinite sets $I$. However we show that $\bigoplus$ and $\prod$ are always
isomorphic on a suitable subcategory of C^I which is isomorphic to C^I but is
not a full subcategory. For the module category case we provide a different
proof to display an interesting connection to the notion of Frobenius corings.
