The Clowder Project

In detail, the left tensor product of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ is the pointed set $\webleft (X\lhd Y,\webleft [x_{0}\webright ]\webright )$ consisting of

• The Underlying Set. The set $X\lhd Y$ defined by
  \begin{align*} X\lhd Y & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\lvert Y\right\rvert \odot X\\ & \cong \bigvee _{y\in Y}\webleft (X,x_{0}\webright ), \end{align*}

  where $\left\lvert Y\right\rvert $ denotes the underlying set of $\webleft (Y,y_{0}\webright )$;

• The Underlying Basepoint. The point $\webleft [\webleft (y_{0},x_{0}\webright )\webright ]$ of $\bigvee _{y\in Y}\webleft (X,x_{0}\webright )$, which is equal to $\webleft [\webleft (y,x_{0}\webright )\webright ]$ for any $y\in Y$.