 Unit identities derived from the monoidal pentagon and triangle

The associator $a$ and unitors of a monoidal category satisfyFor the first identity, postcompose the pentagon on $X,Y,I,Z$ by $1_X\otimes (1_Y\otimes {\lambda }_Z)$, then use the triangle twice and cancel $a_{X,Y,Z}$. This gives the first equality after tensoring with $1_Z$; set $Z=I$ and cancel using the natural invertible right unitor. Reversing tensor order proves the second identity. Naturality of the left unitor at ${\lambda }_X$ gives ${\lambda }_{I\otimes X}=1_I\otimes {\lambda }_X$; comparing the second identity with the triangle on $I,I,X$ gives $({\lambda }_I\otimes 1_X)=({\rho }_I\otimes 1_X)$ and hence the last identity. These are consequences of the axioms, not extra axioms imported from the monoidal coherence theorem.