In detail, a bilinear morphism of pointed sets from $\webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )$ to $\webleft (Z,z_{0}\webright )$ is a map of sets
\[ f \colon \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright ) \to \webleft (Z,z_{0}\webright ) \]
satisfying the following conditions:12
-
Left Unital Bilinearity. The diagram
commutes, i.e. for each $y\in Y$, we have
\[ f\webleft (x_{0},y\webright ) = z_{0}. \] -
Right Unital Bilinearity. The diagram
commutes, i.e. for each $x\in X$, we have
\[ f\webleft (x,y_{0}\webright ) = z_{0}. \]
1Slogan: The map $f$ is bilinear if it preserves basepoints in each argument.
2Succinctly, $f$ is bilinear if we have
\begin{align*} f\webleft (x_{0},y\webright ) & = z_{0},\\ f\webleft (x,y_{0}\webright ) & = z_{0} \end{align*}
for each $x\in X$ and each $y\in Y$.
