Proof that Day Convolution is Closed

Proof. Proof that Day Convolution is Closed

For $𝑋, 𝐵 : {𝒞︀}^{op} \to 𝒱︀$ , the enriched hom in $[{𝒞︀}^{op}, 𝒱︀]$ is given by the end:

\begin{aligned} \int_{𝑐} ((𝐴 \otimes_{Day} 𝑋) 𝑐 ⊸_{𝒱︀} 𝐵 𝑐) & ≅ \int_{𝑐} ((\int^{𝑢, 𝑣} 𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 𝑐) \\ ≅ \int_{𝑐} \int_{𝑢, 𝑣} ((𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 𝑐) \\ ≅ \int_{𝑢, 𝑣} ((𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} \int_{𝑐} (𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) ⊸_{𝒱︀} 𝐵 𝑐)) \\ ≅ \int_{𝑢, 𝑣} ((𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑣)) \\ ≅ \int_{𝑣} (𝑋 𝑣 ⊸_{𝒱︀} \int_{𝑢} (𝐴 𝑢 ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑣))) \\ ≅ \int_{𝑣} (𝑋 𝑣 ⊸_{𝒱︀} (𝐴 ⊸ 𝐵) 𝑣) . \end{aligned}

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