Lemma 13.2.3. Given a third morphism of adequate triples \(H\colon (C,C_L,C_R) \to (D,D_L,D_R)\), let \(\alpha \colon F \Rightarrow G\) and \(\beta \colon G \Rightarrow H\) be natural transformations satisfying the conditions (1) and (2) from Construction 13.2.1. Then we have natural equivalences \[ \Span ^{\mathrm {R}}(\id _F) \simeq \id _{\Span (F)} \qquadtext {and} \Span ^{\mathrm {R}}(\beta \circ \alpha ) \simeq \Span ^{\mathrm {R}}(\beta ) \circ \Span ^{\mathrm {R}}(\alpha ). \] The construction is also compatible with whiskering: if \(K\colon (B,B_L,B_R) \to (C,C_L,C_R)\) and \(J\colon (D,D_L,D_R) \to (E,E_L,E_R)\) are morphisms of adequate triples, then \[ \Span ^{\mathrm {R}}(\alpha K) \simeq \Span ^{\mathrm {R}}(\alpha ) \circ \Span (K) \qquadtext {and} \Span ^{\mathrm {R}}(J\alpha ) \simeq \Span (J) \circ \Span ^{\mathrm {R}}(\alpha ). \] The dual claims for \(\Span ^{\mathrm {L}}(-)\) also hold.
Proof. Composition and identities of natural transformations are formed using the posets \([n]\) for \(n = 0,2\). Whiskering is given by pre- and postcomposition with morphisms of adequate triples. The claims therefore follow from the naturality in both \([n] \in \simp \) and \((C,C_L,C_R) \in \AdTrip \) of the equivalence \(\Span _{L,R}(C \times [n]) \simeq \Span _{L,R}(C) \times [n]\). □
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