Lemma 13.1.13. Let \((C,C_L,C_R)\) be an adequate triple and consider objects \(X,Y \in C\). Then there is an equivalence \[ \Hom _{\Span _{L,R}(C)}(X,Y) \quad \simeq \quad ((C_L)_{/X})^{\simeq } \times _{C^{\simeq }} ((C_R)_{/Y})^{\simeq }. \]
Proof. Observe that the twisted arrow category \(\Tw ^r([1])\) is the walking span \(\;\pushout \) from Definition 1.4.1. We may then compute: \[ \Map ([1],\Span _{L,R}(C)) \simeq \NSpan _{L,R}(C)_1 \simeq \Hom _{\AdTrip }(\pushout , (C,C_L,C_R)) \simeq \Map ([1],C_L) \times _{\ev _0,C^{\simeq },\ev _0} \Map ([1],C_R). \] The hom anima \(\Hom _{\Span _{L,R}(C)}(X,Y)\) is then given as the fiber over \((X,Y)\) of the map \[ \Map ([1],C_L) \times _{\ev _0,C^{\simeq },\ev _0} \Map ([1],C_R) \xrightarrow {(\ev _1, \ev _1)} C^{\simeq } \times C^{\simeq }, \] which is indeed equivalent to \(((C_L)_{/X})^{\simeq } \times _{C^{\simeq }} ((C_R)_{/Y})^{\simeq }\). □
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