Diagrams characterizing ring characteristic, and in particular field characteristic 1?

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  1. For a given prime $p$, what is the diagram for saying (e.g. in some category where morphisms are field homomorphisms) that a field has characteristic $p$?
  2. Is there a diagram, or the shape of what such a diagram would look like, for $F_{un}$, the field of characteristic 1? More directly, is $F_{un}$ characterized by a diagram?