\$$\def\cat#1{{\mathcal{#1}}}\$$ \$$\def\Cat#1{{\textbf{#1}}}\$$
In order for \$$\Cat{1}\$$ to be a monoidal unit, there are supposed to be isomorphisms \$$\cat{X}\times\Cat{1}\to\cat{X}\$$ and \$$\Cat{1}\times\cat{X}\to\cat{X}\$$, for any \$$\cat{V}\$$-category \$$\cat{X}\$$. What are they?