In MacLane Moerdijk, pag. 393, lemma VII.7.3, the following is stated:
Let $f:\mathcal{E}\rightarrow\widehat{\mathbb{C}}$ be a geometric morphism and $J$ a Grothendieck topology over $\mathbb{C}$, then the following are equivalent:
- $f$ factors through $i:Sh(\mathbb{C},J)\hookrightarrow\widehat{\mathbb{C}}$;
- $f^*y$ maps covering sieves in $\mathbb{C}$ to colimits in $\mathcal{E}$;
- $f^*y$ maps covering sieves in $\mathbb{C}$ to epimorphic families in $\mathcal{E}$.
Then $A:\mathbb{C}\rightarrow\mathcal{E}$ is said to be continuous for $J$ if $A$ sends covering sieves to colimits/epimorphic families.
I really fail to grasp the concept here, aside from the formal statement: is there any intuition for this notion of continuity? What happens in particular when such a topology is subcanonical?