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Question: Can the following basic fact be conveniently cited from the literature?
For
$V$ a (co)complete symmetric closed monoidal category
$\mathbf{C}$ a $V$-enriched and -(co)tensored closed monoidal category
$\mathbf{X}$ a small $V$-enriched category
then the $V$-enriched functor category $Func(\mathbf{X},\mathbf{C})$ is $Func(\mathbf{X},\mathbf{V})$-enriched and -(co)-tensored:
The tensoring is objectwise over $\mathbf{X}$ the tensoring of $\mathbf{C}$ over $\mathbf{V}$
the enrichement (powering) is objectwise an end over hom- (power-) objects into the codomain object out of the tensoring of the domain with the corresponding representable.
This follows via standard end-yoga, I may spell it out in the nLab entry later. But what I’d like to know is if there is an existing textbook or other publication that makes this explicit?
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