A geometric morphism between toposes $(f^* \dashv f_*) : \mathcal{E} \to \mathcal{F}$ is surjective or a geometric surjection if it satisfies the following equivalent criteria:
its inverse image functor $f^*$ is faithful (in contrast to the direct image being full and faithful as for a geometric embedding);
its inverse image functor $f^*$ is conservative;
the components $X \to f_* f^* X$ of the adjunction unit are monomorphisms, for all $X \in \mathcal{F}$;
$f^*$ induces a injective homomorphism of subobject lattices
for all $X \in \mathcal{F}$;
$f^*$ reflects the order on subobjects;
$(f^* \dashv f_*)$ is a comonadic adjunction.
The equivalence of these condition appears for instance as MacLaneMoerdijk, VII 4. lemma 3 and prop. 4.
We discuss the equivalence of these conditions:
The equivalence $(f^* \; faithful) \Leftrightarrow (Id \to f^* f_* \;is \; mono)$ is a general property of adjoint functors (see there).
The implication $(f^* faithful) \Rightarrow (f^* induces\;injection\;on\;subobjects)$ works as follows:
first of all $f^*$ does indeed preserves subobjects: since it respects pullbacks and since monomorphisms are characterized as those morphisms whose domain is stable under pullback along themselves.
To see that $f^*$ induces an injective function on subobjects let $U \hookrightarrow X$ be a subobject with characteristic morphism $char U : X \to \Omega$ and consider the image
of the pullback diagram that exhibits $U$ as a subobject. Since $f^*$ preserves pullbacks, this is still a pullback diagram.
If now $U \leq \tilde U$ but $f^* (U) = f^*(\tilde U)$ then both corresponding pullback diagrams are sent by $f^*$ to the same such diagram. By faithfulness this implies that also
commutes, and hence that also $\tilde U \subset U$, so that in fact $\tilde U \simeq U$.
Next we consider the implication ($f^*$ induces injection on subobjects) $\Rightarrow$ ($f^*$ is conservative).
Assume $f^* (X \stackrel{\phi}{\to} X')$ is an isomorphism. We have to show that then $\phi$ is an isomorphism. Consider the image factorization $X \to im(\phi) \hookrightarrow X'$. Since $f^*$ preserves pushouts and pullbacks, it preserves epis and monos and so takes this to the image factorization
of $f^* \phi$, where now the second morphism is an iso, because $f^* \phi$ is assumed to be an iso. By the assumption that $f^*$ is injective on subobjects it follows that also $im \phi \simeq X'$ and thus that $\phi$ is an epimorphism.
It remains to show that $\phi$ is also a monomorphism. For that it is sufficient to show that in the pullback square
we have $X \times_{X'} X \simeq X$. Write $\Delta : X \to X \times_{X'} X$ for the diagonal and let
be its image factorization. Doing the same for $f^* \phi$, which we have seen is a monomorphism, and using that $f^*$ preserves the pullback, we get
Now using again the assumption that $f^*$ is injective on subobjects, this implies $im \Delta_\phi = X \times_{X'} X$ and hence that $\phi$ is a monomorphism.
(…)
The statement about the comonadic adjunction we discuss below as prop. .
For $T : \mathcal{E} \to \mathcal{E}$ a left exact comonad the cofree algebra functor
to the topos of coalgebras is a geometric surjection.
By the discussion at topos of coalgebras the inverse image is the forgetful functor to the underlying $\mathcal{E}$-objects. This is clearly a faithful functor.
Up to equivalence, every geometric surjection is of this form.
This appears for instance as (MacLaneMoerdijk, VII 4., prop 4).
With observation we only need to show that if $f : \mathcal{E} \to \mathcal{F}$ is surjective, then there is $T$ such that
For this, take $T := f^* f_*$. This is a left exact functor by definition of geometric morphism. By assumption on $f$ and using the equivalent definition of def. we have that $f^*$ is a conservative functor. This means that the conditions of the monadicity theorem are met, so so $f^*$ is a comonadic functor.
For more on this see geometric surjection/embedding factorization . Also at monadic descent.
Trivially, any connected geometric morphism is surjective.
For $f : X \to Y$ a continuous function between topological spaces and $(f^* \dashv f_*) : Sh(X) \to Sh(Y)$ the corresponding geometric morphism of sheaf toposes: if $f$ is surjective then $(f^* \dashv f_*)$ is a surjective geometric morphism, conversely, if $(f^* \dashv f_*)$ is a surjective geometric morphism and $Y$ a $T_1$-space then $f$ is surjective.
For a proof see e.g MacLane-Moerdijk, p.367. A similar result holds for injective functions and geometric embeddings but there $T_0$ suffices as a separation requirement on $X$.
geometric surjection
Peter Johnstone, Topos Theory , Academic Press New York 1977. (Available as Dover reprint 2014; Section 4.1)
Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994. (Section VII.4)