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What I'm leaving outside of this comment, since I've alrady ridiculed myself in front of people who actually work with toposes and know this topic better than I do, is the following:

• mention accessible and locally presentable categories. Categories of sheaves are locally presentable elementary topoi. These are very nice categories that albeit being large can be described by a set (sometimes a group is infinite, but can be generated by a finite set; sometimes a category is a proper class, but there is a set of objects generating it).
• Work "on a relative base": sets are no more special than another topos, they're only easier to handle. Most of topos theory works the same way if you fix a "base" topos once and for all, say $\cal S$, and study the 2-category of "toposes over $\cal S$".
• Since a topos is a generalized space, it turns out that this procedure is no different from studying the category of space-maps over a prescribed base $X$ as opposed to the category of all maps with variable codomain. $Set$ being the terminal object of $\bf Topoi$, ${\bf Topoi}/Set$ is merely $\bf Topoi$.

What I'm leaving outside of this comment, since I've alrady ridiculed myself in front of people who actually work with topoi and know this topic better than I do, is the following:

• mention accessible and locally presentable categories. Categories of sheaves are locally presentable elementary topoi. These are very nice categories that albeit being large can be described by a set (sometimes a group is infinite, but can be generated by a finite set; sometimes a category is a proper class, but there is a set of objects generating it).
• Work "on a relative base": sets are no more special than another topos, they're only easier to handle. Most of topos theory works the same way if you fix a "base" topos once and for all, say $\cal S$, and study the 2-category of "topoi over $\cal S$".
• Since a topos is a generalized space, it turns out that this procedure is no different from studying the category of space-maps over a prescribed base $X$ as opposed to the category of all maps with variable codomain. $Set$ being the terminal object of $\bf Topoi$, ${\bf Topoi}/Set$ is merely $\bf Topoi$.

To be more precise, then, categories are places, in each of which we can interpret different kinds of logics (if you like categorical thinking, you better familiarize with the idea that there is plenty of different logics, in the same way -and with the same continuous spectrum of nuances in flavour and aroma- that there is plenty of different coffee beans; also, more than often the debate about the best coffee leads to the same war of religion of the debate about the best logic).

To be more precise, then, categories are places, in each of which we can interpret different kinds of logics (if you like categorical thinking, you better familiarize with the idea that there is plenty of different logics, in the same way -and with the same continuous spectrum of nuances in flavour and aroma- that there is plenty of different coffee beans; also, more than often the debate about the best coffee leads to the same war of religion the debate about the best logic does).

To be more precise, then, categories are places, in each of which we can interpret different kinds of logics (if you like categorical thinking, you better familiarize with the idea that there is plenty of different logics, in the same way -and with the same continuous spectrum of nuances in flavour and aroma- that there is plenty of different coffee machines).

To be more precise, then, categories are places, in each of which we can interpret different kinds of logics (if you like categorical thinking, you better familiarize with the idea that there is plenty of different logics, in the same way -and with the same continuous spectrum of nuances in flavour and aroma- that there is plenty of different coffee beans; also, more than often the debate about the best coffee leads to the same war of religion of the debate about the best logic).