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{-# OPTIONS --without-K #-}
open import Base
open import Homotopy.PullbackDef
module Homotopy.PullbackIsPullback {i} (d : pullback-diag i) where
open pullback-diag d
import Homotopy.PullbackUP as PullbackUP
open PullbackUP d (λ _ → unit)
pullback-cocone : cocone (pullback d)
pullback-cocone = (pullback.a d , pullback.b d , pullback.h d)
factor-pullback : (E : Set i) (cocone E (E pullback d))
factor-pullback E (top→A , top→B , h) x = (top→A x , top→B x , h x)
pullback-is-pullback : is-pullback (pullback d) pullback-cocone
pullback-is-pullback E = iso-is-eq _
(factor-pullback E)
(λ y refl _)
(λ f refl _)
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