# Are fully faithful functors stable under pullback?

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.

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.