Sorry if this is elementary.


Let $C$ be a category, $X$ an object in $C$ and let $S(X)$ denote the poset of subobjects of $X$.


According to the [nlab entry](https://ncatlab.org/nlab/show/subobject#the_poset_of_subobjects), if $C$ has all limits and co-limits, so does $S(X)$. It's a little unclear to me why this is true, and in particular how to construct a limit in $S(X)$ using limits in $C$.


For example, in the category $Mod\_R$ of modules over a ring $R$, $S(X)$ is the poset of submodules of $X$, and the limit and colimit are respectively the intersection and "sum" of submodules. It's not clear to me immediately how these relate to limits and colimits in $Mod\_R$.


Hence the title of the question:


How do (co)limits in poset of subobjects relate to (co)limits in ambient category?



How do (co)limits in posets of subobjects relate to (co)limits in ambient category?
To summarize what's been said in comments:


* If $C$ has small limits, then so does $S(X)$: they are inherited from $C/X$, which inherits them from $C$ by adding $X$ as a terminal object in the diagram being taken a limit of.
* In particular, therefore, if $C$ is well-powered, so that $S(X)$ is small, then it is a small complete lattice and hence also has all colimits, constructed by the adjoint functor theorem. However, they are not *a priori* related to colimits in $C$, and not well-behaved (e.g. not stable under pullback).
* If $C$ has a factorization system where the right class is the subobject inclusions, then joins in $S(X)$ can be constructed as the images of coproducts in $C$.
* If $C$ is a pretopos, then finite joins in $S(X)$ can be constructed as pushouts under intersections, as described at the nLab page.


