Proposition 4.3.11 (Universal property of finite animae). Let \(\An ^{\fin }\) denote the smallest full subcategory of \(\An \) which contains the point and is closed under finite colimits. For every \(\infty \)-category \(D\) with finite limits, evaluation at the point induces an equivalence \[ \ev _*\colon \Fun ^{\lex }((\An ^{\fin })\catop ,D) \iso D. \]
Proof. We construct an inverse. Given \(Y\in \Sp ^{(2)}(C)\) with structure maps \(\tau _k\colon Y_k \iso \Omega ^2Y_{k+1}\), define a spectrum \(QY\) by \[ (QY)_{2k}:=Y_k \qquadtext { and } \qquad (QY)_{2k+1}:=\Omega Y_{k+1}. \] The structure maps of \(QY\) are \(\tau _k\colon Y_k\to \Omega ^2Y_{k+1}=\Omega (QY)_{2k+1}\) in even degrees and the identity map \(\Omega Y_{k+1}\to \Omega Y_{k+1}=\Omega (QY)_{2k+2}\) in odd degrees.
It is immediate that \(EQ=\id \). Conversely, for \(X\in \Sp (C)\) there is a natural isomorphism \(X\to QEX\) which is the identity in even degrees and the structure map \(\sigma ^X_{2k+1}\colon X_{2k+1}\to \Omega X_{2k+2}\) in odd degrees. The compatibility with the structure maps follows directly from the definition of \(\widetilde {\sigma }^X_k\). □
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