Let $\mathcal{C}$ be a finite $k$-linear abelian category, and $\operatorname{Vec}$ be the category of finite dimensional vector spaces over $k$. Let $F_1,\ F_2:\ \mathcal{C}\rightarrow \operatorname{Vec}$ be two $k$-linear exact faithful functors. Then $F_1\otimes F_2$: $\mathcal{C}\times\mathcal{C}\rightarrow\operatorname{Vec}$ is a functor, sending $(X,Y)$ to $F(X)\otimes_kF(Y)$ and $(f,g)$ to $f\otimes g$.
We denote $\operatorname{End}(F)$ as the algebra of natural transformations from $F$ to $F$. To understand the coalgebra structure of $\operatorname{End}(F)$ (when $\mathcal{C}$ is tensor catgeory), an important step is understanding the follow isomorphism:
\begin{align}
\Phi:\operatorname{End}(F_1)\otimes\operatorname{End}(F_2)&\longrightarrow\operatorname{End}(F_1\otimes F_2)\\
\alpha\otimes\beta&\longmapsto \Phi(\alpha\otimes\beta)_{(X,Y)}\!=\!\alpha_X\otimes\beta_Y\!\!: F_1(X)\otimes F_2(Y)\rightarrow F_1(X)\otimes F_2(Y)
\end{align}
It is easy to show that $\Phi$ is a well-defined linear map.(and an algebra map when $F_1=F_2$)
My question is: how to prove that $\Phi$ is isomorphic? I cannot do both injectivity and surjectivity.
Best Answer
Do you find a flaw in this argument?
Fact 0. $\hom(V,W)\cong V^\lor\otimes W$ ($V^\lor$=the dual of $V$).
Fact 1. $A \otimes-$ (tensoring with $A$) is a continuous functor, because it has $A^\lor\otimes-$ (tensoring with the dual of $A$) as left adjoint: $$ \hom(X,A\otimes Y)\cong X^\lor\otimes A^{\lor\lor}\otimes Y\cong \hom(A^\lor \otimes X,Y)$$
Fact 2. For functors $H,K : {\cal C}\to Vect$, $$ Nat(H,K) \cong \int_X Vect(HX,KX)$$ so in particular $End(F)\cong \int_XVect(FX,FX)$.
But now: $$\begin{align*} End(F)\otimes End(G) &\cong \int_X Vect(FX,FX)\otimes \int_YVect(GY,GY) \tag{fact2}\\ &\cong\int_X\int_Y\Big(Vect(FX,FX)\otimes Vect(GY,GY)\Big)\tag{fact1}\\ &\cong\int_X\int_Y(FX)^\lor\otimes FX\otimes (GY)^\lor \otimes GY\tag{fact0}\\ &\cong\int_X\int_Y(FX)^\lor\otimes (GY)^\lor\otimes FX \otimes GY\tag{sym-$\otimes$}\\ &\cong\int_X\int_Y(FX \otimes GY)^\lor\otimes FX \otimes GY\tag{$\lor$-$\otimes$}\\ &\cong\int_{XY}(FX \otimes GY)^\lor\otimes FX \otimes GY\tag{Fubini}\\ &\cong\int_{XY}Vect(FX \otimes GY, FX \otimes GY)\tag{fact0}\\ &\cong End(F\otimes G)\tag{fact2}\\ \end{align*}$$