0NY0
Proof. By Theorem 4.7 QC ( X × Y X ) ≃ QC ( X ) ⊗ QC ( Y ) QC ( X ) \qc(X\times_{Y}X)\simeq\qc(X)\otimes_{\qc(Y)}\qc(X) u = Δ ∗ π ∗ : QC ( Y ) → QC ( X × Y X ) u=\Delta_{*}\pi^{*}:\qc(Y)\to\qc(X\times_{Y}X) τ = π ∗ Δ ∗ : QC ( X × Y X ) → QC ( Y ) \tau=\pi_{*}\Delta^{*}:\qc(X\times_{Y}X)\to\qc(Y) Δ : X → X × Y X \Delta:X\to X\times_{Y}X
QC ( X ) \textstyle{\qc(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces} u ⊗ id \scriptstyle{u\otimes\operatorname{id}} QC ( X ) ⊗ QC ( Y ) QC ( X ) ⊗ QC ( Y ) QC ( X ) \textstyle{\qc(X)\otimes_{\qc(Y)}\qc(X)\otimes_{\qc(Y)}\qc(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces} id ⊗ τ \scriptstyle{\operatorname{id}\otimes\tau} QC ( X ) \textstyle{\qc(X)}
The argument is a chase in the following diagram (with Cartesian square):
X \textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces X\ignorespaces\ignorespaces\ignorespaces\ignorespaces} Δ \scriptstyle{\Delta} Δ \scriptstyle{\Delta} X × Y X \textstyle{X\times_{Y}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} π 1 \scriptstyle{\pi_{1}} id 1 × Δ 23 \scriptstyle{{\rm id}_{1}\times\Delta_{23}} X \textstyle{X} X \textstyle{X} X × Y X \textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces X\times_{Y}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces} π 2 \scriptstyle{\pi_{2}} Δ 12 × id 3 \scriptstyle{\Delta_{12}\times{\rm id}_{3}} X × Y X × Y X \textstyle{X\times_{Y}X\times_{Y}X}
Applying base change and identities for compositions, we have equivalences of functors
( id ⊗ τ ) ∘ ( u ⊗ id ) \displaystyle(\operatorname{id}\otimes\tau)\circ(u\otimes\operatorname{id}) = \displaystyle= π 1 ∗ ( id 1 × Δ 23 ) ∗ ( Δ 12 × id 3 ) ∗ π 2 ∗ \displaystyle\pi_{1*}({\rm id}_{1}\times\Delta_{23})^{*}(\Delta_{12}\times{\rm id}_{3})_{*}\pi^{*}_{2}
≃ \displaystyle\simeq π 1 ∗ Δ ∗ Δ ∗ π ∗ 2 \displaystyle\pi_{1*}\Delta_{*}\Delta^{*}\pi^{*}_{2}
≃ \displaystyle\simeq id QC ( X ) \displaystyle{\rm id}_{\qc(X)}
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