2 typo

What kind of categories $C$ have the property that each slice category $C/c$ is a topos? Obviously, topoi have this property, but, the converse is not true. An example is the category $EtTop$ of topological spaces and only local homeomorphisms. $EtTop/X \cong Sh(X)$, but $EtTop$ is very far from being a topos! It doesn't have an initial or final object, nor pushouts...

Are there other of these "locally a topos" categories which are not topoi? Have they been studied before?

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# Locally a topos

What kind of categories $C$ have the property that each slice category $C/c$ is a topos? Obviously, topoi have this property, but, the converse is not true. An example is the category $EtTop$ of topological spaces and only homeomorphisms. $EtTop/X \cong Sh(X)$, but $EtTop$ is very far from being a topos! It doesn't have an initial or final object, nor pushouts...

Are there other of these "locally a topos" categories which are not topoi? Have they been studied before?