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 <a href="https://qchu.wordpress.com/2015/10/26/dualizable-objects-and-morphisms/">this blog post</a>, 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.)