Let's say if $u:C\to D$ is a functor between small categories. I am wondering why the nerve $$N(c):N(C)\to N(D)$$ becomes an inner Kan fibration?
An inner Kan fibration is a morphism with the right lifting property with respect to the inclusion of inner horns.
I feel like it should be obvious so I must have missed something. Thank you so much for any help!