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Let $A$, $B$ and $C$ be locally small categories, let $I:A\to C$ be a fully faithful functor and let $F:B\to C$ be any functor. Is the pullback $F^*I: A \times_C B\to B$ still fully faithful?

If yes, is there an underlying orthogonal factorization system?

Any reference would also be welcome.

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  • $\begingroup$ Of course one can also just check it directly with the definition. $\endgroup$ Commented Jun 2, 2021 at 23:32

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There is a bijective-on-objects/fully faithful orthogonal factorisation system on Cat. Hence, as a right class of a factorisation system, fully faithful functors are closed under pullback in Cat.

For another reference, see also Riehl's notes on Factorisation Systems.

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  • $\begingroup$ +1 - really slick answer $\endgroup$ Commented Jul 20, 2020 at 21:53

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