Is the category of toposes cocomplete ? Hello.
[Edits between brackets.]
Does the [1-]category of [elementary] toposes [with logical morphisms] admit any [1-]colimits ?
[By colimit I mean initial object in the category of outgoing cocones. I'm afraid I don't know what pseudo-colimits, bicolimits or strict 2-limits are, but I can say that I'm not interested by any 2-categorical aspect of my question (at least, for now), just by the 1-categorical one.]
Any reference would be greatly appreciated.
Thanks for any answer and please forgive me for my pityful english.
 A: $\mathbf{Log}$, the category of elementary topoi and logical morphisms which preserve everything on the nose is cocomplete; in

Eduardo J Dubuc, GM Kelly - A presentation of topoi as algebraic relative to categories or graphs - Journal of Algebra (website)

this category $\mathbf{Log}$ is shown to be the category of algebras of a finitary monad on $\mathbf{Cat}$; it looks like you can even get this over $\mathbf{Grph}$, like the presentation of cartesian closed categories monadic over graphs in Lambek-Scott intro to higher-order categorical logic. This implies then that it is cocomplete, see the nLab page on colimits in categories of algebras, for example. 
For the sake of completeness, a nice account (including explicit constructions) of 2-limits in $\mathbf{Log}$ as a locally groupoidal 2-category (1-cells the standard notion of logical morphism, 2-cells between them restricted to be isomorphisms) is given in chapter III of Steve Awodey's PhD thesis, 

S Awodey - Logic in topoi: functorial semantics for higher-order logic - available from his website

A: You should consult section B3 Of the Elephant by Johnstone.  My understanding is that the 2-category of Grothendieck toposes (over sets) has all small colimits. This holds more generally for bounded toposes over a topos with natural numbers object.
