Proposition 23.6.6 (Hom animae in functor categories). Let \(F,G \colon C \to D\) be functors. Then we have the following end-formula for the anima of natural transformations from \(F\) to \(G\): \[ \Nat (F,G) \quad \simeq \quad \int _{x \in C} \Hom _D(F(x), G(x)). \] Here the right-hand side is the end of the composite functor \(C\catop \times C \xrightarrow {F\catop \times G} D\catop \times D \xrightarrow {\Hom _D} \An \).
Proof. The end imposes precisely the naturality conditions on a pointwise family of morphisms \(F(x)\to G(x)\). The rigorous identification, including all higher coherence data, is proved in Reference ? of [Cisinski et al. (2026)]. □
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