How would one specify the category of all (elementary) toposes? There seems to be not that much about it on the internet: all I could find is this MO question where it is given the name $\mathbf{Log}$.
What would be the objects of $\mathbf{Log}$?
Elementary toposes $\mathscr{C}$
Pairs $(\mathscr{C},\Omega)$ where $\mathscr{C}$ is a topos with subobject classifier $\Omega$
I realized I have the same dilemma for many other categories: should we, in $\mathbf{Top}$, write $(X,\tau)$ instead of $X$ for its objects (similar for $(G,\cdot)$ instead of $G$ in $\mathbf{Grp}$)?
The problem with this is that the subobject clasifier is unique up to isomorphism so it somehow comes implicitly when saying $\mathscr{C}$ is a topos (I'm not sure it is unique up to unique ismorphism or what role does $\;\top:1\to\Omega\;$ play)
What would be the morphisms of $\mathbf{Log}$? Any functor? Should they preserve anything (the clasiffier, the character of a monic, etc)?
Last question: Is the category of all toposes itself a topos? If so what are the $\Omega$ and $\top:1\to\Omega$ and what would we get to know about the logic carried away there?
Thanks