Let $(C, \otimes, \alpha, 1, l, r)$ be a monoidal category. Show that $l_{1 \otimes A} = id_1 \otimes l_A$ and $r_{A \otimes 1} = r_A \otimes id_1$.

Question

Let $(C, \otimes, \alpha, 1, l, r)$ be a monoidal category. I would like to show that

for any object $A$ in $C$ the equalities $l_{1 \otimes A} = id_1 \otimes l_A$ and $r_{A \otimes 1} = r_A \otimes id_1$ hold true.

A hint tells me that one should use functoriality of the left and right unitor $l$ and $r$, but I do not understand what this means in this case. I am probably missing the definition of a functorial isomorphism. Can someone help me to prove the above identities ?

Thanks for your help.

Answer 1

$l_A : 1\otimes A\cong A$ is natural in $A$. That means the following is a naturality square: $$\require{AMScd} \begin{CD} 1\otimes (1\otimes A) @>id_1\otimes l_A>> 1\otimes A \\ @Vl_{1\otimes A}VV @VVl_AV \\ 1\otimes A @>>l_A> A \end{CD}$$ That is, $l_A\circ (id_1\otimes l_A) = l_A\circ l_{1\otimes A}$ to which we post-compose with $l_A^{-1}$ to get $id_1\otimes l_A = l_{1\otimes A}$.