Semicocartesian Monoidal Category

A Monoidal Category is semicocartesian if the unit is an Initial Object.

Properties

• Every tensor $X\otimes Y$ in a semicocartesian monoidal category $\mathcal{V}$ comes equipped with maps $X\xrightarrow{i_{1}}X\otimes Y\xleftarrow{i_{2}}Y$ given by

  	$$\displaystyle i_{1}=mathrm{id}{ifx..lse\par i}\otimes\mbox{!`}\circ\rho_{X}$$
  	$$\displaystyle i_{2}=\mbox{!`}\otimes mathrm{id}{ifx..lse\par i}\circ\lambda_{X}$$

• If there is a coduplication map $\nabla:X\otimes X\to X$ such that $\nabla\circ i_{1}=mathrm{id}{ifx..lse\par i}$ and $\nabla\circ i_{2}=mathrm{id}{ifx..lse\par i}$, then $X\otimes Y$ is a Coproduct.