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 $f^* (X \stackrel{\simeq}{\to} X')$ 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. 2.
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 1 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. 1 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.
For $f : X \to Y$ a continuous function between topological spaces and $(f^* \dashv f_*) : Sh(X) \to Sh(Y)$ the corresponding geometric morphisms of sheaf toposes, $f$ is a surjection precisely if $(f^* \dashv f_*)$ is a surjective geometric morphism.
geometric surjection
Section VII. 4. of