Hom$_*$ commuting with filtered products

In a sufficiently rich category, such as a category of R-modules, and a given infinite cardinal $κ$, we examine classes $\Cal H^κ_*$ of objects M, such that the following natural monomorphism is an isomorphism: $$\prod_{i\in I}}^κ\hbox{\rm Hom}\,(M, A_i)\cong \hbox{\rm Hom}\,(M, \prod_{i\in I}}^κ A_i),$$ for every family of objects $\{A_i, i\in I\}$ ($\prod^κ$ denotes the subproduct of all the vectors with support $<κ$).

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