  We systematically investigate morphisms and equivalences of toposes from
multiple points of view. We establish a dual adjunction between morphisms and
comorphisms of sites, introduce the notion of weak morphism of toposes and
characterize the functors which induce such morphisms. In particular, we
examine continuous comorphism of sites and show that this class of comorphisms
notably includes all fibrations as well as morphisms of fibrations. We also
establish a characterization theorem for essential geometric morphisms and
locally connected morphisms in terms of continuous functors, and a relative
version of the comprehensive factorization of a functor. Then we prove a
general theorem providing necessary and sufficient explicit conditions for a
morphism of sites to induce an equivalence of toposes. This stems from a
detailed analysis of arrows in Grothendieck toposes and denseness conditions,
which yields results of independent interest. We also derive site
characterizations of the property of a geometric morphism to be an inclusion
(resp. a surjection, hyperconnected, localic), as well as site-level
descriptions of the surjection-inclusion and hyperconnected-localic
factorizations of a geometric morphism.
