A strange ring category

This is useful is you would like a Hopf algebra to be a group object in the category of algebras. If $A$ is a Hopf algebra, then the antipode map $S: A \to A$, is an antiendomorphism of $A$. You can see this from the example of group algebras: if $G$ is a group, and $k[G]$ is the group algebra, then $k[G]$ has a Hopf algebra structure where $S: g \mapsto g^{-1}$ and we have $(gh)^{-1} = h^{-1} g^{-1}$.

I don't know that it's worth defining a category just to fix this issue. I have seen people spend far more time bothered by this issue than it is worth when learning Hopf algebras, so maybe this would have helped them.

I also think that this category is kind of pathological, because it probably lacks limits and colimits. Perhaps a little bit more natural is the category with rings as objects and where the morphisms are either homomorphisms or antihomomorphisms, in the sense that we take the disjoint union of these sets (thus, a homomorphism of, say, commutative rings is distinguished from the corresponding antihomomorphism). Let me explain how this category can be constructed from a general method.

Let $\mathcal{C}$ be a category. Let $D : \mathcal{C} \to \mathcal{C}$ be a functor with $D \circ D = \mathrm{id}$. For $\mathcal{C}=\mathsf{Mon},\,\mathsf{Grp},\,\mathsf{Ring}$ or $\mathsf{Cat}$ we take $D$ to be the "opposite-object" functor. Define a new category $\mathcal{C}_D$ as follows: The objects are the objects of $\mathcal{C}$. For objects $x,y$ we define $$\hom_{\mathcal{C}_D}(x,y) = \hom_{\mathcal{C}}(x,y) \sqcup \hom_{\mathcal{C}}(x,D(y)).$$ The identity of an object in $\mathcal{C}_D$ is the identity in $\mathcal{C}$. The composition of $x \to y$ with $y \to z$ in $\mathcal{C}_D$ is the one in $\mathcal{C}$. The composition of $x \to y$ with $y \to D(z)$ is the composition $x \to D(z)$ in $\mathcal{C}$, regarded as an element in the second part of $\hom_{\mathcal{C}_D}(x,z)$. The composition of $x \to D(y)$ with $y \to z$ is the composition $x \to D(y) \to D(z)$ in $\mathcal{C}$, again regarded as an element in the second part of $\hom_{\mathcal{C}_D}(x,z)$. Finally, the composition of $x \to D(y)$ with $y \to D(z)$ is the composition $x \to D(y) \to D(D(z)) = z$, now regarded as an element in the first part of $\hom_{\mathcal{C}_D}(x,z)$. One checks that $\mathcal{C}_D$ is, indeed, a category.