Let $(H, \mu, \eta, \Delta, \epsilon, S)$ be a Hopf algebra with $S: H \to H$ denoting the antipode. By definition, $S$ is the convolution inverse of $1: H \to H$ in $\operatorname{End}(H)$, with the convolution product $f*g$ given by the composition $$H \xrightarrow{\Delta} H \otimes H \xrightarrow{f \otimes g} H \otimes H \xrightarrow{\mu} H$$ It is commonly known that $S$ is an antihomomorphism ($S(xy) = S(y)S(x)$), but the only proof I am aware of is via direct computation (like Prop III.3.3 in Kassel's Quantum Groups).
While I can spend time on understanding the proof, I wonder if there is a more categorical proof of this. In the end, a Hopf algebra makes sense in any symmetric (or braided) monoidal category, and $S$ being an antihomomorphism simply translates to asking the equality $$S \circ \mu = \mu \circ \tau \circ (S \otimes S): H \otimes H \to H$$ (where $\tau: H \otimes H \to H \otimes H$ is the braiding). Thus, my speculation is that a purely diagrammatic proof should exist, unless the statement is not true for a general symmetric monoidal category (in which case I would like to know what makes vector spaces special). Does anyone have a reference?
