See [here](https://mathoverflow.net/questions/303581/is-the-category-2-vect-monoidal-closed) for the notation.


I'm trying to do this by myself but everytime I approach the problem I end up buried under computations.



> 
> What is a pair of adjoint 1-cells in the 2-category of vector spaces, where
> 
> 
> * 0-cells are natural numbers $\langle N\rangle,\langle M\rangle...$
> * 1-cells $\langle N\rangle\to \langle M\rangle $ are strictly-positive-integer valued matrices of size $M\times N$
> * 2-cells $A\to B$ are matrices whose entries are matrices, whose $(i,j)$ entry is a matrix having size $A\_{ij}\times B\_{ij}$.
> 
> 
> ?
> 
> 
> 



Adjoints in the 2-category of 2-vector spaces
As in the previous question, I continue to suggest that everything is much cleaner if you think in terms of bimodules. So we'll ask the more general question: in the 2-category $\text{Bim}(k)$ of $k$-algebras, $k$-bimodules, and bimodule homomorphisms, what is an adjoint pair? 


The answer is worked out in [this blog post](https://qchu.wordpress.com/2015/10/26/dualizable-objects-and-morphisms/), although I'll warn you that in the convention of that blog post composition of 1-morphisms is written in diagrammatic order, which means concretely that the composition of an $(A, B)$-bimodule $M$ and a $(B, C)$-bimodule $N$, in that order, is $M \otimes\_B N$. Adopting the usual composition convention switches the meaning of "left" and "right," but this convention makes things work out very nicely as follows. 



> 
> **Proposition:** An $(A, B)$-bimodule $M$ has a left adjoint iff it is finitely presented projective as a left $A$-module, in which case its left adjoint is $\text{Hom}\_A(M, A)$. It has a right adjoint iff it is finitely presented projective as a right $B$-module, in which case its right adjoint is $\text{Hom}\_B(M, B)$. 
> 
> 
> 


KV is the special case where we only consider algebras of the form $k^n$ for $k$ a field and $(k^n, k^m)$-bimodules, which concretely can be thought of as $n \times m$ matrices of $k$-vector spaces, where all of the vector spaces are finite-dimensional. These are all finitely presented projective on both sides, and so we conclude that every morphism of KV 2-vector spaces has both a left and a right adjoint, both of which I believe are concretely given by sending a matrix $V\_{ij}$ of vector spaces to the transposed and dualized matrix $V\_{ji}^{\ast}$, but I haven't checked this in detail. 


(Note that because I'm thinking about 1-morphisms as matrices of vector spaces instead of taking a skeleton and identifying vector spaces with their dimensions, when I write down taking the dual I'm making some implicit claims about functoriality and 2-morphisms that I don't need to keep explicit track of. It's actually more work to work with the skeleton because then I need to distinguish a vector space from its dual by talking a lot more explicitly about 2-morphisms, and I sort of have to conjure the 2-morphisms out of thin air rather than letting them naturally arise out of e.g. the dual pairing.) 


