Throughout, $\mathsf C_1,\mathsf C_2$ are small categories and $\mathscr V$ is a bicomplete symmetric monoidal closed category.
I'm trying to understand some of the basic theory of group representations. This blog post by Qiaochu Yuan points out the internal hom on the category of representations acts as a "categorified inner product". This makes the picture much more conceptual for me.
However, now I'm struggling to understand how to see the fact that simple representations of a product of groups come from the outer tensor product of simple representations of their factors.
I tried to understand this answer, but I'm still missing something. Given two representations $F_1:\mathsf C_1\to \mathscr V$ and $F_2:\mathsf C_2\to \mathscr V$ the assignment $F_1,F_2\mapsto F_1\boxtimes F_2;\eta,\mu\mapsto \eta\boxtimes \mu$ given by applying the tensor product of $\mathscr V$ pointwise yields a functor $$[\mathsf C_1,\mathscr V]\times [\mathsf C_2,\mathscr V]\overset{\boxtimes}{\to}[\mathsf C_1\times \mathsf C_2,\mathscr V].$$ The latter inherits a symmetric monoidal closed structure from $\mathscr V$ and so we have the composite $$([\mathsf C_1,\mathscr V\times [\mathsf C_2,\mathscr V])^\text{op}\times [\mathsf C_1\times \mathsf C_2,\mathscr V]\overset{\boxtimes^\text{op}\times 1}{\longrightarrow}[\mathsf C_1\times \mathsf C_2,\mathscr V]^\text{op}\times [\mathsf C_1\times \mathsf C_2,\mathscr V]\overset{\mathsf{hom}}{\longrightarrow}\mathscr V.$$ By symmetry of products and Cartesian-closedness of $\mathsf{Cat}$ we obtain a functor $$\mathrm H:[\mathsf C_1\times \mathsf C_2,\mathscr V]\longrightarrow[([\mathsf C_1,\mathscr V]\times [\mathsf C_2,\mathscr V])^\text{op},\mathscr V]$$ defined on objects by $$\mathrm H(F)= \mathsf{hom}(-\boxtimes=,F):([\mathsf C_1,\mathscr V]\times [\mathsf C_2,\mathscr V])^\text{op}\to \mathscr V.$$ The claim is then that this functor is representable when $F$ is simple. Then, by enriched Yoneda for $\mathsf C_1\times \mathsf C_2$ we somehow deduce that a simple $F$ has the form $F_1\boxtimes F_2$ for simple $F_1,F_2$. Why can we do this?
I also tried to understand this answer to the same question. Namely that given a factor $F_1$ we can get the other one using some sort of $\mathsf{hom}(F_1,F)$. This is a little confusing to because I read it as $ \left\langle F, F_1 \right\rangle$ and that shouldn't provide information about $F_2$, and also because I don't see how to give this internal hom the structure of a representation of $\mathsf{C_2}$. On the other hand, the previous question sort of hints at this by using an "internal tensor-hom adjunction" on $H(F)$, but I also don't see how to make any sense of that.
If anyone could illuminate these things to me, that'd be great.