TITLE: How do (co)limits in posets of subobjects relate to (co)limits in ambient category?
QUESTION [5 upvotes]: 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, 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?

REPLY [4 votes]: 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.