7.1.1 Left Bilinear Morphisms of Pointed Sets
Let $(X,x_{0})$, $(Y,y_{0})$, and $(Z,z_{0})$ be pointed sets.
A left bilinear morphism of pointed sets from $(X\times Y,(x_{0},y_{0}))$ to $(Z,z_{0})$ is a map of sets
\[ f \colon X\times Y \to Z \]
satisfying the following condition:
- (★)
Left Unital Bilinearity. The diagram commutes, i.e. for each $y\in Y$, we have
\[ f(x_{0},y) = z_{0}. \]
The set of left bilinear morphisms of pointed sets from $(X\times Y,(x_{0},y_{0}))$ to $(Z,z_{0})$ is the set $\smash {\operatorname {\mathrm{Hom}}^{\otimes ,\mathrm{L}}_{\mathsf{Sets}_{*}}(X\times Y,Z)}$ defined by
\[ \operatorname {\mathrm{Hom}}^{\otimes ,\mathrm{L}}_{\mathsf{Sets}_{*}}(X\times Y,Z) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ f\in \operatorname {\mathrm{Hom}}_{\mathsf{Sets}}(X\times Y,Z)\ \middle |\ \text{$f$ is left bilinear}\right\} . \]
