An elementary topos is a category with finite limits, exponential objects, and a subobject classifier. Here a quote from Leinster's An informal introduction to topos theory:
More spectacularly, the axioms imply that every topos has finite colimits. This
can be proved by the following very elegant strategy, due to Paré (1974). For
every topos $E$, we have the contravariant power set functor $P = Ω^{(−)}
: E^{
op} → E$ .
It can be shown that $P$ is monadic. But monadic functors create limits, and $E$
has finite limits. Hence $E^{
op}$ has finite limits; that is, $E$ has finite colimits.
As far as I now, the fact that $P$ creates limits means that $A$ is the limit of a diagram of objects $A_i$ in $E^{op}$ if and only if $PA$ is the limit of all $P(A_i)$ in $E$. Since $E$ has finite limits, there is a limit $L$ of all $P(A_i)$ (provided the diagram $(A_i)$ is finite). Hence if $L$ has the form $PA$, then we can conclude that $(A_i)$ has a limit, namely $A$, in $E^{op}$, i.e., a colimit in $E$.
Question: How does one conclude that $E$ has all finite colimits? The problem I see is that $L$ isn't necessarily of the form $PA$.
Best Answer
Your definition of what it means for a funtor to create limits is incomplete: creating limits means not only preserving and reflecting, but also that the a diagram in the domain has a limit whenever its image in the codomain does. It is in this sense that monadic functors create limits.