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(Caveat: I come from set theory rather than category theory and know only a little about ETCS, but I believe that the ETCS.)

The answer to your question is no. The basic reason is that even the models of set theory themselves can differ vastly. If $M$ is a model of ZFC, then the category $Set^M$, which is Set as interpreted in $M$, will be a model of ETCS. But if ZFC is consistent, then the models of set theory $M$ are diverse. For example, some have CH and others have $\neg CH$, and furthermore, by the incompleteness theorem, they can satisfy different arithmetic statements. Such statements show up in the category $Set^M$, since I believe that every arithmetic statement (first order statement about natural numbers) has a translation into the formal language of ETCS. So in general they these categories are not elementary equivalent in the language of ETCS. In particular, the natural numbers objects of such categories will not in general be isomorphic, you and so there can expect be no nice functors between such modelsthe categories. For example, as by the Lowenheim-Skolem theorem, some can models of set theory will be countable and others will have an uncountable set of natural numbersobject, and others notwith a different theory, and all kinds of crazy thingsthese aspects will prevent their corresponding Set categories from being equivalent as categories or from having nice functors. In general, it will not be possible to map the natural number object from one to the other in any nice way.

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I don't know much only a little about ETCS, but I believe that the answer will be to your question is no. The basic reason is that even the models of set theory themselves can differ vastly. If $M$ is a model of ZFC, then the category $Set^M$, which is Set as interpreted in $M$, will be a model of ETCS. But if ZFC is consistent, then the models of set theory $M$ are diverse. For example, some have CH and others have $\neg CH$, and furthermore, by the incompleteness theorem, they can satisfy different arithmetic statements. Such statements show up in the category $Set^M$, since I believe that every arithmetic statement (first order statement about natural numbers) has a translation into the formal language of ETCS. So in general they are not elementary equivalent. In general, you can expect no nice functors between such models, as some can have uncountable natural numbers object, and others not, and all kinds of crazy things. In general, it will not be possible to map the natural number object from one to the other in any nice way.

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I don't know much about ETCS, but I believe that the answer will be no. The basic reason is that even the models of set theory themselves can differ vastly. If $M$ is a model of ZFC, then the category $Set^M$, which is Set as interpreted in $M$, will be a model of ETCS. But if ZFC is consistent, then the models of set theory $M$ are diverse. For example, some have CH and others have $\neg CH$, and furthermore, by the incompleteness theorem, they can satisfy different arithmetic statements. Such statements show up in the category $Set^M$, since I believe that every arithmetic statement (first order statement about natural numbers) has a translation into the formal language of ETCS. So in general they are not elementary equivalent. In general, you can expect no nice functors between such models, as some can have uncountable natural numbers object, and others not, and all kinds of crazy things. In general, it will not be possible to map the natural number object from one to the other in any nice way.