TITLE: Are dagger categories truly evil?
QUESTION [60 upvotes]: Recall that a dagger category is a category equipped with an involution $*:Hom(x,y)\to Hom(y,x)$ that satisfies $f^{**}=f$ and $f^* g^*=(gf)^*$. A prominent example of a dagger category is the category of Hilbert spaces and continuous linear maps.

Now, dagger categories are evil!
For example, here's a quote from the Nlab:

"Note that regarded as an extra structure on categories, a †-structure is evil, since it imposes equations on objects."

This is not just a little thing. It's a fundamental philosophical problem with category theory: category theory does not seem to be able to cope with this rather important category-theory-related notion which are dagger categories.
Also, because of this buit-in evilness, there's a whole bunch of very familiar concepts that one cannot apply to dagger categories without rethinking everything very, very carefully from the beginning: equivalences, limits, colimits, adjoint functors, duals, algebra objects, etc...
Here's another quote:

"It is possible that this problem will force a change in thinking in either the concept of the principle of equivalence or our thinking in quantum theory."

and yet another one:

"Often concepts violating the principle of equivalence (like the concept of “strict monoidal category”) have equivalence-invariant counterparts (like the concept of “monoidal category”). But in this particular case there appears to be no known way to express the idea without equations between objects."

Now here's my question:
Question: Can one improve the above "there appears to be no known way to express the idea without equations between objects" to "there is no way to express the idea without equations between objects"?
I expect dagger categories to be truly evil (in some yet-to-be-defined technical sense of "evil"). Having a proof of that fact might be very informative, and would at least force us to define the term "evil" at a mathematical level of precision.

REPLY [9 votes]: Here's a notion of undirected category, which is equivalent to the notion of dagger category. It's a bit cumbersome to work with, but the main point is to underscore Qiaochu Yuan and Manuel Bärenz's points that daggers are not really a categorical structure, and dagger categories are their own thing.
An undirected category consists of a pair of maps $s,t:C_1 \to C_0$, with constant $\mathrm{id} : C_0 \to C_1$ and instead of one composition operation for composing arrows tip to tail, there are four composition operations for composing in any direction (composition is in diagrammatic order):
${}_t;_s : Hom(A,B) \times Hom(B,C) \to Hom(A,C)$
${}_s;_s : Hom(B,A) \times Hom(B,C) \to Hom(A,C)$
${}_t;_t : Hom(A,B) \times Hom(C,B) \to Hom(A,C)$
${}_s;_t : Hom(B,A) \times Hom(C,B) \to Hom(A,C)$
Satisfying the following equations ($a, b$ and $c$ are variables in $s$ and $t$, and $-s = t$, $-t = s$)
$f = f {}_t;_a \mathrm{id} = \mathrm{id} {}_a;_s f$ (usual identity)
$f {}_s;_a \mathrm{id} = \mathrm{id} {}_a;_t f =: f^\dagger$
$(f {}_a;_b g) {}_s;_c h = (g {}_b;_a f) {}_t;_c h = g {}_b;_s  (f {}_{-a};_c h) = g {}_b;_t  (h {}_{c};_{-a} f)$ (the middle one includes usual associativity).
It's not hard to show that given a dagger category, you can get an undirected category by defining the compositions as $f {}_s;_t g := f^\dagger ; g^\dagger$ and the like, with the same identity, and given an undirected category, define a dagger as in the second line of equations above, and to show that these are inverses of each other.
Then there is also the fact that algebras of the path monad on the category of undirected graphs is exactly an undirected/dagger category, just as an algebra of the path monad on the category of directed graphs is a category.