The tensor of $\webleft (X,x_{0}\webright )$ by $A$[1] is the pointed set[2] $A\odot \webleft (X,x_{0}\webright )$ 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 $\webleft (K,k_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
