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Jan 23, 2020 at 10:48 comment added Martin Brandenburg I think it is not fair to wait for an answer which answers all three questions. I suggest to open a new thread for the whole monad thing (which is, in fact, not necessary for Kelly's tensor product).
Jan 21, 2020 at 20:29 comment added Ivan Di Liberti I think I understood, but I will not accept this answer, hopefully will come with a more explanatory answer. This answers Q1, but does not make Kelly more readable at all. That was the main point of my question.
Jan 21, 2020 at 19:47 comment added Giorgio Mossa I've made an edit. It should address your doubts.
Jan 21, 2020 at 19:46 history edited Giorgio Mossa CC BY-SA 4.0
Made an edit to address a comment.
Jan 21, 2020 at 19:20 comment added Ivan Di Liberti This is more or less a copy-paste of Kelly's words, but I frankly do not get them. $A \otimes B$ should be a $V$-category with $\mathcal{F}$-colimits, not a theory, nor a sketch. I am probably missing something incredibly simple, but I do not understand Kelly's words.
Jan 21, 2020 at 17:59 history answered Giorgio Mossa CC BY-SA 4.0