= Unit identities derived from the monoidal pentagon and triangle
{title2=$\lambda_I=\rho_I$}
The <associator> $a$ and <unitors> of a <monoidal category> satisfy
$$
\rho_{X\otimes Y}=(1_X\otimes\rho_Y)a_{X,Y,I},\qquad
\lambda_{X\otimes Y}a_{I,X,Y}=\lambda_X\otimes1_Y,\qquad
\lambda_I=\rho_I.
$$
For 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 $(\lambda_I\otimes1_X)=(\rho_I\otimes1_X)$ and hence the last identity. These are consequences of the axioms, not extra axioms imported from the <monoidal coherence theorem>.
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