Are there conditions to be fulfilled by a site with terminal object and products to be a subsite of a cartesian closed site?

20 October 2019 0 5K Report

A site (C, J) is given such that category C has a terminal object and products.

I am looking for conditions to be fulfilled by (C, J) ensuring the existence of a cartesian closed site (E, K) such that C is a subcategory of E and the Grothendieck topology J is induced by K.

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