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.

$\mathrm{𝐒𝐩𝐅𝐢}$ is the category of spaces with filters: an object is a pair $(X,ℱ)$, $X$ a compact Hausdorff space and $ℱ$ a filter of dense open subsets of $X$. A morphism $f(Y,𝒢)\to (X,ℱ)$ is a continuous function $fY\to X$ for which $f^{-1}\left(F\right)\in 𝒢$ whenever $F\in ℱ$. This category arises naturally from considerations in ordered algebra, e.g., Boolean algebra, lattice-ordered groups and rings, and from considerations in general topology, e.g., the theory of the absolute and other covers, locales, and frames, though we shall specifically address only one of these...

Conditions which imply Morita equivalences of functor categories are described. As an application a Dold-Kan type theorem for functors defined on a category associated to associative algebras with one-side units is proved.

Let ${(e_{\beta }:𝐐\to Y_{\beta })}_{\beta \in \mathbf{\text{Ord}}}$ be the large source of epimorphisms in the category $\mathbf{\text{Ury}}$ of Urysohn spaces constructed in [2]. A sink ${(g_{\beta }:Y_{\beta }\to X)}_{\beta \in \mathbf{\text{Ord}}}$ is called natural, if $g_{\beta }\circ e_{\beta }=g_{{\beta }^{\text{'}}}\circ e_{{\beta }^{\text{'}}}$ for all $\beta ,{\beta }^{\text{'}}\in \mathbf{\text{Ord}}$. In this paper natural sinks are characterized. As a result it is shown that $\mathbf{\text{Ury}}$ permits no $(Epi,ℳ)$-factorization structure for arbitrary (large) sources.

In order to facilitate a natural choice for morphisms created by the (left or right) lifting property as used in the definition of weak factorization systems, the notion of natural weak factorization system in the category $𝒦$ is introduced, as a pair (comonad, monad) over ${𝒦}^{\mathbf{2}}$. The link with existing notions in terms of morphism classes is given via the respective Eilenberg–Moore categories.