Do faithful representations of a $C^*$-algebra $A$ forms a category? If yes, what are the morphisms?
My guess is that if we take natural transformations arising due to unitary operators as morphism, then faithful representations of a $C^*$-algebra forms a category. Suppose $\pi_1\colon A\to B(H_1)$ is faithful representation and $\pi_2(x)=U\pi_1(x) U^*$, where $U^*U=I=UU^*$. Then $\pi_2$ must be faithful. Am I doing this right?
Any time you have a class with an equivalence relation $\sim$, you have a category where the objects are the elements of the class, and for any two objects $x,y$ there is a morphism $x\to y$ if and only if $x\sim y$. Moreover, this category will be a "groupoid", in the sense that it is a category (not necessarily small) in which every morphism is an isomorphism.
Since unitary equivalence defines an equivalence relation of representations, which preserves faithfulness, you get a category.