Let $\webleft (X,x_{0}\webright )$ be a pointed set and let $A$ be a set.
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Functoriality. The assignments $A,\webleft (X,x_{0}\webright ),\webleft (A,\webleft (X,x_{0}\webright )\webright )$ define functors
\[ \begin{array}{ccc} A\odot -\colon \mkern -15mu & \mathsf{Sets}\mathrlap {{}_{*}} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ -\odot X\colon \mkern -15mu & \mathsf{Sets} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ -_{1}\odot -_{2}\colon \mkern -15mu & \mathsf{Sets}\times \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*}. \end{array} \]
In particular, given:
- A map of sets $f\colon A\to B$;
- A pointed map $\phi \colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright )$;
\[ f\odot \phi \colon A\odot X\to B\odot Y \]is given by
\[ \webleft [f\odot \phi \webright ]\webleft (a\odot x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (a\webright )\odot \phi \webleft (x\webright ) \]for each $a\odot x\in A\odot X$.
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Adjointness I. We have an adjunction witnessed by a bijection
\[ \mathsf{Sets}_{*}\webleft (A\odot X,K\webright )\cong \mathsf{Sets}\webleft (A,\mathsf{Sets}_{*}\webleft (X,K\webright )\webright ), \]natural in $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and $X,Y\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
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Adjointness II. We have an adjunctions witnessed by a bijection
\[ \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (A\odot X,Y\webright )\cong \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X,A\pitchfork Y\webright ), \]natural in $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and $X,Y\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
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As a Weighted Colimit. We have
\[ A\odot X\cong \text{colim}^{\webleft [A\webright ]}\webleft (X\webright ), \]
where in the right hand side we write:
- $A$ for the functor $A\colon \text{pt}\to \mathsf{Sets}$ picking $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$;
- $X$ for the functor $X\colon \text{pt}\to \mathsf{Sets}_{*}$ picking $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
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Iterated Tensors. We have an isomorphism of pointed sets
\[ A\odot \webleft (B\odot X\webright )\cong \webleft (A\times B\webright )\odot X, \]
natural in $A,B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
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Interaction With Homs. We have a natural isomorphism
\[ \mathsf{Sets}_{*}\webleft (A\odot X,-\webright )\cong A\pitchfork \mathsf{Sets}_{*}\webleft (X,-\webright ). \]
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The Tensor Evaluation Map. For each $X,Y\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, we have a map
\[ \mathrm{ev}^{\odot }_{X,Y}\colon \mathsf{Sets}_{*}\webleft (X,Y\webright )\odot X\to Y, \]
natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, and given by
\[ \mathrm{ev}^{\odot }_{X,Y}\webleft (f\odot x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (x\webright ) \]for each $f\odot x\in \mathsf{Sets}_{*}\webleft (X,Y\webright )\odot X$.
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The Tensor Coevaluation Map. For each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $X\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, we have a map
\[ \mathrm{coev}^{\odot }_{A,X}\colon A\to \mathsf{Sets}_{*}\webleft (X,A\odot X\webright ), \]
natural in $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and $X\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, and given by
\[ \mathrm{coev}^{\odot }_{A,X}\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[x\mapsto a\odot x]\mspace {-3mu}] \]for each $a\in A$.
