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adjusted the Idea-paragraph
in particular I added a line that $V$-strong functors make sense between $V$-tensored $V$-enriched categories
and hence deleted this sentence further down:
The first thing to notice about (covariant) tensorial strengths is that they attach to a functor from a monoidal category to itself, say $T: V \to V$. (The concept doesn’t make much immediate sense if $T$ is a functor between different monoidal categories.)
added pointer to:
Kruna S. Ratkovic, Tensorial Strength, Section 3 of: Morita theory in enriched context (2012) [arXiv:1302.2774, hal:tel-00785301]
Tarmo Uustalu, Strong functors, Strong monads, in: Containers for effects and contexts, Lecture notes (2015) [pdf]
Dylan McDermott, Tarmo Uustalu, Section 3 of: What Makes a Strong Monad?, EPTCS 360 (2022) 113-133 [arXiv:2207.00851, doi:10.4204/EPTCS.360.6]
I have given the paragraphs alluding to some work by C. S. Peirce an Examples-environment (here) but it remains a little mysterious without further details or references.
In the course of this, I have taken the liberty of deleting this lead-in paragraph:
There’s rather a lot more one could say about strengths, and I may come back to more of that later, but I would like to say that strengths are kind of a trade secret. The first mathematician I know of who intuitively grasped strength was C.S. Peirce!
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