### Simultaneously reflective and coreflective subcategories of presheaves.

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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.

Commutative semigroups satisfying the equation $2x+y=2x$ and having only two $G$-invariant congruences for an automorphism group $G$ are considered. Some classes of these semigroups are characterized and some other examples are constructed.

If $R$ is a prime ring such that $R$ is not completely reducible and the additive group $R(+)$ is not complete, then $R$ is slender.

Left distributive quasitrivial groupoids are completely described and those of them which are subdirectly irreducible are found. There are also determined all left distributive algebras $A=A(*,\circ )$ such that $A(*)$ is a quasitrivial groupoid.

This short note is a continuation of and and its purpose is to show that every simple zeropotent paramedial groupoid containing at least three elements is strongly balanced in the sense of .

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.

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