For every module $M$ we have a natural monomorphism $$\Psi :\coprod _{i\in I}{\mathrm{H}om}_R(M,A_i)\to {\mathrm{H}om}_R\left(M,\coprod _{i\in I}{A}_i\right)$$ and we focus our attention on the case when $\Psi$ is also an epimorphism. Some other colimits are also considered.

For every module M we have a natural monomorphism  $\Phi :{\coprod }_{i\in I}Ho{m}_R(A_i,M)\to Ho{m}_R({\prod }_{i\in I}{A}_i,M)$ and we focus attention on the case when Φ is also an epimorphism. The corresponding modules M depend on thickness of the cardinal number card(I). Some other limits are also considered.