Are fully faithful functors stable under pullback?

Question

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.

Answer 1

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.