John Chrysostom
Do you see the clear type? There, he says, “outside the camp” (Heb 13:11), and here, “outside the gates.” Just as the sacrifices offered for sins were a kind of type and were burned outside the camp, so Jesus, who offered Himself for our sins, suffered outside the gates. Therefore we too must imitate Him who suffered for us and be outside the world, or rather, outside worldly deeds. [...] Then the Apostle explains the typological sacrifice and turns the discourse to the archetype: “For the bodies of those animals whose blood is brought into the sanctuary by the high priest for sin are burned outside the camp. Therefore Jesus also, that He might sanctify the people by His own Blood, suffered outside the gates.” Consequently, the first was a type of the latter.
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Thus Christ fulfilled everything by suffering outside the gates. Here the Apostle also expresses that Christ suffered voluntarily. He proves that those sacrifices were not established without purpose, but were a kind of type. He also proves that the divine economy itself was not accomplished without sufferings, but that Christ’s Blood was carried up into heaven.