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The universal properties of products and coproducts "amount" to the statements

$$ \hom(\coprod_i X_i , Y) = \prod_i \hom(X_i, Y) \quad \text{and} \quad \hom(X,\prod_i Y_i) = \prod_i \hom(X,Y_i)$$

for any category and any objects for which the (co)product is defined. I am wondering if there is any general statement about the other two cases: are there any similar formulas that describe $\hom(X,\coprod_i Y_i)$ and $\hom(\prod_i X_i , Y)$? I don't expect a general formula to exist, but maybe one can say something at least for abelian categories?

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    $\begingroup$ In general categories, I think you should not expect any meaningful answer, here. In abelian categories you have finite biproducts, so if the products/coproducts are finite, then of course you get the same formulas. This leaves the infinite product in an abelian category case -- I once again doubt you'll get anything nice, but take that with a grain of salt. $\endgroup$ – Mees de Vries Feb 25 at 14:36
  • $\begingroup$ In an extensive category, $X$ is a connected object if $\mathsf{Hom}(X,-)$ preserves all coproducts, i.e.the canonical morphism $\coprod_i\mathsf{Hom}(X,Y_i)\to\mathsf{Hom}(X,\coprod_i Y_i)$ is an isomorphism. $\endgroup$ – Derek Elkins Feb 25 at 19:55

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