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This is a question regarding set-theoretic issues in higher category theory. Let $\text{Cat}_{\infty}^{big}(\omega)$ be the $\infty$-category of (not necessarily small) $\infty$-categories with morphisms those functors preserving $\omega$-filtered colimits. Let $\text{Cat}_{\infty}(\omega)$ be the category of essentially small $\infty$-categories with morphisms preserving $\omega$-filtered colimits. My question is the following: Does $\text{Cat}_{\infty}(\omega)\subset \text{Cat}_{\infty}^{big}(\omega)$ preserve $\omega$-filtered colimits? Any useful discussion regarding closely related set theoretic issues in (higher) category theory would be nice.

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