Hi,

In a strict monoidal category, where the associator, left and right unitor are identity morphisms we have the following relations between (string) diagrams:

where $i_x$ and $e_x$ are the unit and counit isomorphisms.

For the first relation for example, the LHS starts by $1\otimes x$ and goes to $x\otimes 1$, but the two are really the same object $x$ since we're in a stirct monidal category, so we go from $x$ to $x$, which is exactly what happens in the RHS.

What I don't understand, is why such equalities are still valid in a weakened monoidal category, where now $x\otimes 1$ and $1\otimes x$ are merely isomorphic.