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This question is based on section 5.2 in Tensor Categories, by Etingof et al. Note also that the question is pretty much in the title and what follows is just some background along with my fruitless thoughts.

Let $\mathcal{C}$ be a tensor category over $k$ and let $(F,J)$ be a fiber functor. We have a unital algebra $H\equiv \mathrm{End}(F)$, multiplication is composition.

We also have an algebra isomorphism \begin{align*} \alpha: \mathrm{End}(F)\otimes \mathrm{End}(F) \to \mathrm{End}(F\otimes F) \end{align*} with components \begin{align*} \alpha(\eta\otimes\zeta)_{X,Y}\equiv \eta_X\otimes\zeta_Y, \end{align*} where $F\otimes F:\mathcal{C}\times\mathcal{C}\to \mathsf{Vec}$ is the obvious functor, $F\otimes F(X,Y)\equiv FX\otimes FY$.

Using this, we define \begin{align*} \Delta : H &\to H\otimes H,\\ \eta &\mapsto \alpha^{-1}(J^{-1}\circ \eta \circ J), \end{align*} where $(J^{-1}\circ \eta \circ J)_{X,Y} = J^{-1}_{X,Y}\circ \eta_{X\otimes Y} \circ J_{X,Y}$, and \begin{align*} \varepsilon:H&\to k,\\ \eta&\mapsto \eta_1. \end{align*}

I don't see why these satisfy the comultiplication and counit axioms. In the book it is stated that this follows from the definition of monoidal functor and fiber functor, so this means we should be able to draw a diagram of some sort.

But elements of $\mathrm{End}(F)\otimes \mathrm{End}(F)$ are not natural transformations anymore, so how to handle them?

For example, to show one way of counit, we want $(\varepsilon\otimes 1)\circ \Delta (\eta) = \eta$. The obvious route would be to show that components agree, so write it out: \begin{align*} [(\varepsilon\otimes 1)\circ \Delta (\eta)]_X &= [(\varepsilon\otimes 1) (\alpha^{-1}(J^{-1}\circ \eta \circ J))]_X \\ &= \sum_{ij} \tilde{\eta}_{ij} (e_i)_1 (e_j)_X \end{align*} where we have chosen a basis $\{e_i\}$ of the vector space $\mathrm{End}(F)$, and wrote $\alpha^{-1}(J^{-1}\circ \eta \circ J) = \sum\tilde{\eta}_{ij}e_i\otimes e_j$. This doesn't show me anything.

And I don't even know where to start if we were to use commutative diagrams to show this.

Any insight is appreciated!

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Let me assume that both the category $\mathcal{C}$ and the functor $F$ are strict, to simplify the exposition. First, some preliminaries. For $F,G$ be two functors from $\mathcal{C}$ to $\mathsf{Vec}$, consider $$\alpha_{F,G}:\mathsf{End}(F)\otimes \mathsf{End}(G)\to \mathsf{End}(F\otimes G),$$ $$\alpha_{F,G}(\eta\otimes\zeta)_{X,Y}=\eta_X\otimes\zeta_Y:F(X)\otimes G(Y)\to F(X)\otimes G(Y),$$ and write $c_J:\mathsf{End}(F)\to \mathsf{End}(F\otimes F)$ for the assignment $$c_J(\eta)_{X,Y}=J_{X,Y}^{-1}\circ\eta_{X\otimes Y}\circ J_{X,Y}.$$ Notice that $\alpha_{F,F}\circ\Delta=c_J$ and that we may also consider, for example, $$c_{F\otimes J}:\mathsf{End}(F\otimes F)\to \mathsf{End}(F\otimes F\otimes F),$$ $$c_{F\otimes J}(\gamma)_{X,Y,Z}=\left(F(X)\otimes J^{-1}_{Y,Z}\right)\circ\gamma_{X,Y\otimes Z}\circ\left(F(X)\otimes J_{Y,Z}\right).$$ A direct check shows that \begin{equation}\label{alphanat}\tag{$*$} \alpha_{F,F\otimes F}\circ\left(\mathsf{End}(F)\otimes c_J\right) = c_{F\otimes J}\circ\alpha_{F,F},\\ \alpha_{F\otimes F,F}\circ\left(c_J\otimes \mathsf{End}(F)\right) = c_{J\otimes F}\circ\alpha_{F,F}. \end{equation} Finally, recall that since $(F,J)$ is monoidal, the hexagonal condition $J\circ(F\otimes J) = J \circ (J\otimes F)$ is satisfied and hence \begin{equation}\label{J}\tag{$\star$} c_{F\otimes J} \circ c_J = c_{J\otimes F}\circ c_J. \end{equation}

Now, let me show the coassociativity condition, for example. Set $H:=\mathsf{End}(F)$ and compute

\begin{align*} \alpha_{F,F \otimes F} & \circ\left(H \otimes \alpha_{F,F}\right) \circ \left(H \otimes \Delta\right) \circ \Delta = \alpha_{F,F \otimes F} \circ \left(H\otimes c_J\right) \circ \Delta \\ & \stackrel{\eqref{alphanat}}{=} c_{F\otimes J} \circ \alpha_{F,F} \circ \Delta = c_{F\otimes J} \circ c_J \stackrel{\eqref{J}}{=} c_{J\otimes F} \circ c_J \\ & = c_{J\otimes F} \circ \alpha_{F,F} \circ \Delta \stackrel{\eqref{alphanat}}{=} \alpha_{F\otimes F,F} \circ \left(c_{J}\otimes H\right) \circ \Delta \\ & = \alpha_{F\otimes F,F} \circ \left(\alpha_{F,F}\otimes H\right) \circ \left(\Delta\otimes H\right) \circ \Delta. \end{align*} Since $\alpha_{F\otimes F,F} \circ \left(\alpha_{F,F}\otimes H\right)=\alpha_{F,F\otimes F} \circ \left(H \otimes \alpha_{F,F}\right)$ and they are natural isomorphism, coassociativity follows.

In general, the idea is to use the fact that $\alpha$ provides a natural isomorphism to check conditions in a suitable space of natural transformations.

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