Since $A\cong \mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}_{a\in A}\text{pt}$ and $\otimes _{\mathsf{Sets}}$ preserves colimits in each variable, we have
\begin{align*} A\otimes _{\mathsf{Sets}}B & \cong \webleft (\coprod _{a\in A}\text{pt}\webright )\otimes _{\mathsf{Sets}}B\\ & \cong \coprod _{a\in A}\webleft (\text{pt}\otimes _{\mathsf{Sets}}B\webright )\\ & \cong \coprod _{a\in A}B\\ & \cong A\times B, \end{align*}
naturally in $B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, where we have used that $\text{pt}$ is the monoidal unit for $\otimes _{\mathsf{Sets}}$. Thus $A\otimes _{\mathsf{Sets}}-\cong A\times -$ for each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.