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Let $(M, \otimes, I)$ be a symmetric monoidal category and let $(A, B, \eta, \epsilon)$ be a dual pair in $M$. Consider maps

$i_{A}: Hom(A, A) \rightarrow Hom(A \otimes B, A \otimes B)$,

$i_{B}: Hom(B, B) \rightarrow Hom(A \otimes B, A \otimes B)$

given by tensoring with the identity. One can show that these maps are injective. (As $i_{A}$ followed by composition with unit establishes the well-known bijection $Hom(A, A) \simeq Hom(I, A \otimes B)$ and analogously for $i_{B}$).

My question concerns characterizing the image of these maps. Observe that we have a canonical bijection $Hom(A, A) \simeq Hom(B, B)$ and it might seem under first impression that under this bijection the two maps should coincide. However, this is false.

I first worked this out in possibly the simplest non-trivial case, namely when $M = Vect_{k}$ equipped with the tensor product of vector spaces. Let $A = V$, $B = V^{*}$ and $A \otimes B \simeq End(V)$. By explicit computations it turns out that in this case

$i_{V}: Hom(V, V) \rightarrow Hom(End(V), End(V))$

$i_{V}(f)(g) = (f \circ g)$

and

$i_{V^{*}}: Hom(V, V) \simeq Hom(V^{*}, V^{*}) \rightarrow Hom(End(V), End(V))$

$i_{V^{*}}(f)(g) = (g \circ f)$

Hence, one can make a naive conjecture as follows:

A map $f \in Hom(A \otimes B, A \otimes B)$ lies in $im(i_{A})$ if and only if it commutes with all $g \in im(i_{B})$.

Is the "conjecture" true? If not, are there any other categorical characterizations of the images? Ideally I would want to present $im(i_{A})$ as fixed points of some group (or monoid) action, but maybe that's asking too much.

Another reason to at least consider the conjecture is that the calculation for vector spaces can be redone categorically and in general $M$ we can prove that for $f \in Hom(A, A)$, $i_{A}(f)$ acts by postcomposition with $f$ on "elements" $g \in Hom(I, A \otimes B) \simeq Hom(A, A)$. Therefore we would be done if $Hom(I, -)$ was a faithful functor, but this can easily fail (look at the category of sets with disjoint union as tensor product). However, it would be enough for $Hom(I, -)$ to be faithful on the full subcategory spanned by dualizable objects.

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