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
\[ \begin{array}{ccc} A\pitchfork -\colon \mkern -15mu & \mathsf{Sets}\mathrlap {{}_{*}} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ -\pitchfork X\colon \mkern -15mu & \mathsf{Sets}^{\mathrlap {\mathsf{op}}} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ -_{1}\pitchfork -_{2}\colon \mkern -15mu & \mathsf{Sets}^{\mathsf{op}}\times \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*}. \end{array} \]
In particular, given:
- A map of sets $f\colon A\to B$;
- A pointed map $\phi \colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright )$;
\[ f\odot \phi \colon A\pitchfork X\to B\pitchfork Y \]is given by
\[ \webleft [f\odot \phi \webright ]\webleft (\webleft [\webleft (x_{a}\webright )_{a\in A}\webright ]\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (\phi \webleft (x_{f\webleft (a\webright )}\webright )\webright )_{a\in A}\webright ] \]for each $\webleft [\webleft (x_{a}\webright )_{a\in A}\webright ]\in A\pitchfork X$.
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Adjointness I. We have an adjunction witnessed by a bijection
\[ \mathsf{Sets}^{\mathsf{op}}_{*}\webleft (A\pitchfork X,K\webright )\cong \mathsf{Sets}\webleft (A,\mathsf{Sets}_{*}\webleft (K,X\webright )\webright ), \]i.e. by a bijection
\[ \mathsf{Sets}_{*}\webleft (K,A\pitchfork X\webright )\cong \mathsf{Sets}\webleft (A,\mathsf{Sets}_{*}\webleft (K,X\webright )\webright ), \]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 Limit. We have
\[ A\pitchfork X\cong \text{lim}^{\webleft [A\webright ]}\webleft (X\webright ), \]
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 Cotensors. We have an isomorphism of pointed sets
\[ A\pitchfork \webleft (B\pitchfork X\webright )\cong \webleft (A\times B\webright )\pitchfork X, \]
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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Commutativity With Homs. We have natural isomorphisms
\begin{align*} A\pitchfork \mathsf{Sets}_{*}\webleft (X,-\webright ) & \cong \mathsf{Sets}_{*}\webleft (A\odot X,-\webright ),\\ A\pitchfork \mathsf{Sets}_{*}\webleft (-,Y\webright ) & \cong \mathsf{Sets}_{*}\webleft (-,A\pitchfork Y\webright ). \end{align*}
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The Cotensor Evaluation Map. For each $X,Y\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, we have a map
\[ \mathrm{ev}^{\pitchfork }_{X,Y}\colon X\to \mathsf{Sets}_{*}\webleft (X,Y\webright )\pitchfork Y, \]
natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, and given by
\[ \mathrm{ev}^{\pitchfork }_{X,Y}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (f\webleft (x\webright )\webright )_{f\in \mathsf{Sets}_{*}\webleft (X,Y\webright )}\webright ] \]for each $x\in X$.
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The Cotensor Coevaluation Map. For each $X\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$ and each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, we have a map
\[ \mathrm{coev}^{\pitchfork }_{A,X}\colon A\to \mathsf{Sets}_{*}\webleft (A\pitchfork X,X\webright ), \]
natural in $X\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$ and $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, and given by
\[ \mathrm{coev}^{\pitchfork }_{A,X}\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[\webleft [\webleft (x_{b}\webright )_{b\in A}\webright ]\mapsto x_{a}]\mspace {-3mu}] \]for each $a\in A$.
