In topos theory, there are many generalizations of topological concepts. For example, open, closed, proper and etale morphisms between toposes. However, there are also such analogous concepts in algebraic geometry.

My question is that do these concepts actually coincide? I mean, for example, a proper morphism of schemes actually induces a proper geometric morphism between some toposes induced by schemes (like etale topos)?

Although I mensioned about only morphisms, I want to know such analogous concepts in topology and algebraic geometry which are coincide at the level of toposes.

Proper Maps of Toposesby I. Moerdijk and J.J.C. Vermeulen, Memoirs of AMS, 2000. $\endgroup$