I know that God despises sin... Would this statement be there because it is talking about the shame that comes with sin? Or is it talking about how Christ was feeling when he was on the cross? He did it for his Father, so wouldn't he have peace about what he did? Or why would he feel shame in dying on the cross?
RV1909 - 2 Puestos los ojos en al autor y consumador de la fe, en Jesús; el cual, habiéndole sido propuesto gozo, sufrió la cruz, menospreciando la vergüenza, y sentóse á la diestra del trono de Dios.
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public execution is not simply rendering judgment, or to serve as an example, but to also shame the person (downgrade his honor/reputation) before everyone else.
Thus Christ's crucifixion was a public execution to also shame his person, him especially being deity. The creator to suffer and die like his creation in a most humiliating way; a public spectacle.
Heb_6:6.. because they are nailing the Son of God to the cross again by rejecting him, holding him up to public shame.