The category of fields, denoted Field, is the full subcategory of CRing whose objects are fields.
In Field, the category of fields, there are no initial or terminal objects. However, in the subcategory of fields of fixed characteristic, the prime field is an initial object.
[...] A curious aspect of the category of fields is that every morphism is a monomorphism
In Field we have no initial or terminal objects but this is not true if we are in the subcategory of fields of fixed characteristic because we can say that a prime field is an initial object.
I want to understand better this situation: before we have neither initial/terminal object nor zero object but then we can 'extract' an initial object if we move into a "subcategory of fields of fixed characteristic". What makes this possible?
So.. what structures/morphism we have in $\text{field}\rightarrow\text{subcategory of fields of fixed characteristic}$ to return a initial object as prime field ?
I don't understand well the purpose of this 'subcategorification'