In their book Tensor Categories Etingof, Gelaki, Nikshych and Ostrik give a different definition of a (strong) monoidal functor. The difference is that they do not set the isomorphism $F(1) \cong 1$ as a part of the data, but rather impose a condition on the pair $(F\colon C\to C', (J_{X,Y}\colon F(X)\otimes F(Y)\to F(X\otimes Y))_{X,Y \in C})$ that would be a monoidal functor that some isomorphism $F(1) \cong 1$ exists. They then define a canonical isomorphism $F(1)\cong 1$ by the following diagram:
The exercise is then given to prove that, for this canonical isomorphism, the following diagrams commute:
My trouble is the second diagram. Clearly, we first should tensor diagram defining $\phi$ with $F(X)$ and $1_{F(X)}$ and appeal to functoriality, but I don't see how we can get from $1_{F(X)}\otimes (\phi \otimes 1_1)$ to $1_{F(X)}\otimes \phi$.
Best Answer
As Max already pointed out, you may start with tensoring the diagram (2.24) defining $\varphi$ with $F(X)$. So you get the commutative diagram (1):
Notice, that (1) looks like the desired diagram (2.26), tensored with $F(1)$: the diagram (2):
Now we are going to show that (2) is isomorphic to (1), which immediately implies that (2.26) is commutative. To prove that (2) is isomorphic to (1) we should prove the commutativity of the certain cube with (1) and (2) as opposite faces. To indicate faces I will use the Rubik's Cube notation. Define the B-side as (1) and F-side as (2). Set the U-side as the following diagram:
which is commutative by the triangle diagram. The L-side is simply associativity, so I will not draw it. The D-side is following:
It is commutative by the monoidal structure axiom. The R-side is automatically defined, let's prove its commutativity. Indeed,
$$
J_{X\otimes 1,1}^{-1}\circ F(a_{X,1,1}^{-1})\circ J_{X,1\otimes 1}\circ(\text{id}_{F(X)}\otimes^{\wr}F(\ell_1^{-1}))=
J_{X\otimes 1,1}^{-1}\circ F(a_{X,1,1}^{-1})\circ F(\text{id}_X\otimes\ell_1^{-1})\circ J_{X,1}=
$$
$$
J_{X\otimes 1,1}^{-1}\circ F(a_{X,1,1}^{-1}\circ(\text{id}_X\otimes\ell_1^{-1}))\circ J_{X,1}=
J_{X\otimes 1,1}^{-1}\circ F(r^{-1}_X\otimes\text{id}_1)\circ J_{X,1}=
F(r^{-1}_X)\otimes^{\wr}\text{id}_{F(1)},
$$
where the first and fourth equalities are the naturality of $J$, the second is the functoriality of $F$ and the third is the triangle diagram for the domain monoidal category.