K, thx. If there is any doubt with notation, something big I don't explain because I forgot it is non-trivial, just write some terms, I'll add hyperlinks to the post like in wikipedia, where needed.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)