The left tensor product of pointed sets satisfies the following natural bijection:
\[ \mathsf{Sets}_{*}\webleft (X\lhd Y,Z\webright )\cong \textup{Hom}^{\otimes ,\mathrm{L}}_{\mathsf{Sets}_{*}}\webleft (X\times Y,Z\webright ). \]
That is to say, the following data are in natural bijection:
- Pointed maps $f\colon X\lhd Y\to Z$.
- Maps of sets $f\colon X\times Y\to Z$ satisfying $f\webleft (x_{0},y\webright )=z_{0}$ for each $y\in Y$.
