Item 8 (00D9) — The Clowder Project

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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 $a\in A$.

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