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let $\mathcal{A}$ be a 2-category, consider:

$$ \mathcal{A}(W \otimes_i A, X) \simeq_i \mathcal{C}at(W, \mathcal{A}(A, X)) \;\;\; i = 1, \: 2, \: 3. $$

where $W$ is a category, and $A$, $X$ are objects of $\mathcal{A}$

$\otimes_1$: $\simeq_1$ equivalence pseudonatural on $X$

$\otimes_2$: $\simeq_2$ isomorphism pseudonatural on $X$

$\otimes_3$: $\simeq_3$ isomorphism 2-natural on $X$

Then,

$W \otimes_1 A$ is a $bitensor$, and it is a pseudofunctor on A.

$W \otimes_3 A$ is a $tensor$ (in the sense of enriched categories over $\mathcal{C}at$), and it is a 2-functor on A.

But, what about $W \otimes_2 A$ ?.

It has be considered somewhere ?. Does it have a name (I suggest $pseudotensor$), Is it a 2-functor on A ?. I am not convinced about the suggested name. Actually, $\otimes_3$ is the pseudotensor = tensor because there are no pseudocones involved.

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  • $\begingroup$ I try: consider $A''\to A'\to A$ and the inducted morphisms on the right element (take these vertically), consider $\cong_2$ you get a 2-square diagram (without the left vertical), you can fill in only one way the lef vertical making the diagram commutative. By YOneda LEmma, and becaus the right element is a 2-functor, you get that $W \otimes_2 (-)$ is a 2-functor. What is wrong on mine argument? $\endgroup$
    – Buschi Sergio
    May 30, 2015 at 18:22
  • $\begingroup$ In my above comment, YOneda Lemma connot applied in the usual form. THen the question become: let $\phi: (Y, -)\to (X, -): \mathcal{A}\to Cat$, pseudot., let $f_\phi:= \phi(1_Y)$, how is related $\phi$ and $(f_\phi, -)$ ? We have a (someway) natural $\phi(g)\cong g\circ f_\phi$ for $g\in (Z, Y)$, then given $\psi: (Y, -)\to (Z, -)$ also, we get $f_{\psi\circ \phi}= \psi(f_\phi)\cong f_\psi\circ f_\phi$. THis make $W \otimes_2 (-)$ a pseudofunctor. $\endgroup$
    – Buschi Sergio
    May 30, 2015 at 18:59
  • $\begingroup$ If $\phi(a)_X: (W\otimes A, X)\to (W\otimes A', X)$ for $a: A'\to A$ if the morphism induced as in mine comment(1), is easy (make a cube diagram) that $\phi$ has a natural structure of pseudotransformation on $X$, then comment(2) apply. $\endgroup$
    – Buschi Sergio
    May 30, 2015 at 19:30
  • $\begingroup$ Thank you Sergio, you yourself saw what it was wrong in your first argument. Your second also applies to $\otimes_1$, and the pseudofunctoriality of $\otimes_2$ is just a particular case. The problem remains, can we do better in the case $\otimes_2$ ?. $\endgroup$
    – Eduardo J. Dubuc
    May 31, 2015 at 17:59

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