Theorem is
If φ is a field homomorphism from field F to K then φ is either identical zero or injective.
I know how this come but when we say φ is identical zero then what about multiplicative identity of F. Is it map to zero? If so then we know that multiplicative identity map to multiplicative identity then how is it possible.
On
There are different conventions about ring (and field) homomorphisms. Some people require that the multiplicative identity is mapped to the multiplicative identity, and some don't. In this case you've clearly come across someone who doesn't recuire such a ting.
And yes, when they say that $\varphi$ is identically zero, that means, among other things, that $\varphi(1_F) = 0_K$.
It depends if you postulate field morphisms to conserve the multiplicative unit. If you do, all field morphisms are injective. If you don't, the zero morphism is permitted too.