The tensor of $\webleft (X,x_{0}\webright )$ by $A$1 is the pointed set2 $A\odot \webleft (X,x_{0}\webright )$ satisfying the following universal property:
- We have a bijection
\[ \mathsf{Sets}_{*}\webleft (A\odot X,K\webright )\cong \mathsf{Sets}\webleft (A,\mathsf{Sets}_{*}\webleft (X,K\webright )\webright ), \]
natural in $\webleft (K,k_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
1Further Terminology: Also called the copower of $\webleft (X,x_{0}\webright )$ by $A$.
2Further Notation: Often written $A\odot X$ for simplicity.
