Item 3 (01W4) — The Clowder Project

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

Since $X\cong \bigvee _{x\in X^{-}}S^{0}$ and $\otimes _{\mathsf{Sets}_{*}}$ preserves colimits in each variable, we have
\begin{align*} X\otimes _{\mathsf{Sets}_{*}}Y & \cong \webleft (\bigvee _{x\in X^{-}}S^{0}\webright )\otimes _{\mathsf{Sets}_{*}}Y\\ & \cong \bigvee _{x\in X^{-}}\webleft (S^{0}\otimes _{\mathsf{Sets}_{*}}Y\webright )\\ & \cong \bigvee _{x\in X^{-}}Y\\ & \cong \bigvee _{x\in X^{-}}S^{0}\wedge Y\\ & \cong \webleft (\bigvee _{x\in X^{-}}S^{0}\webright )\wedge Y\\ & \cong X\wedge Y, \end{align*}

naturally in $Y\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, where we have used that $S^{0}$ is the monoidal unit for $\otimes _{\mathsf{Sets}_{*}}$. Thus $X\otimes _{\mathsf{Sets}_{*}}-\cong X\wedge -$ for each $X\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.

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