In any set with structure, we define isomorphisms as being homomorphisms with the extra property of bijection. I'm reading a book about category theory and the definiton is only about bijection. There's no $F(f\circ g) = F(f)\circ F(g)$. Is this implicit and I can't see?
The given definition is talking about homomorphisms between two objects in a category, and what it means for such a thing to be an isomorphism.
From your question, your trouble is that you have a different context in mind and so you misunderstand the definition — you are thinking about what a homomorphism between two categories would be. Such a thing is called a functor, and $F(g \circ f) = F(g) \circ F(f)$ is among the properties required for $F$ to be a functor.