Let $C$ be a category and let $C'$ be the wide subcategory whose maps are projections, that is maps in $C$ which belong to some limiting cone (over a discrete base). Since limiting cones compose, $C'$ is indeed a subcategory of $C$ (containing all isomorphisms of $C$).

For instance, if $C$ is the category of finite sets, a mapping $f:A\to B$ is in $C'$ if and only if all fibers $f^{-1}y$ have the same cardinality. Thus the postal reflection of $C'$ is equivalent to the poset of natural numbers ordered by divisibility. Note that this $C'$ has not finite products (although a terminal set is terminal also in $C'$). Or (dually) one can consider topological spaces obtaining the category of embeddings which are summands.

My question:

has this notion been already considered?

are there interesting instances?

Let $C$ be a category and let $C'$ be the wide subcategory whose maps are projections, that is maps in $C$ which belong to some limiting cone (over a discrete base). Since limiting cones compose, $C'$ is indeed a subcategory of $C$ (containing all isomorphisms of $C$).

For instance, if $C$ is the category of finite sets, a mapping $f:A\to B$ is in $C'$ if and only if all fibers $f^{-1}y$ have the same cardinality. Thus the posetal reflection of $C'$ is equivalent to the poset of natural numbers ordered by divisibility. Note that this $C'$ has not finite products (although a terminal set is terminal also in $C'$). Or (dually) one can consider topological spaces obtaining the category of embeddings which are summands.

My question:

has this notion been already considered?

are there interesting instances?