In this post on bimonoid objects in monoidal categories, certain axioms are presented as "string diagrams", but nowhere can I find how I'm supposed to interpret them. I found somewhere that morphisms are visualized as circles, and objects as lines, which makes sense.
However, I'm not sure how to actually translate them to equations, or which morphisms and objects are even involved. Easiest for the answerer I think would just be to translate the diagrams to algebra, so I can generalize myself what the diagrams mean. (Or you could explain it, if this is easier to you).
A red dot is the multiplication $\mu:B\otimes B\to B$ or unit $\eta:I\to B$ of $B$, while a blue dot is the comultiplication $\nu:B\to B\otimes B$ or counit $\varepsilon:B\to I$ of $B$. A crossing of wires is an application of the symmetry $\tau$ of the monoidal category. A blank space is the (identity morphism of) the unit object $I$ of the ambient category. Parallel wires indicate tensoring.
So for instance the first displayed diagram indicates that $\nu\circ \mu=\mu\otimes \mu\circ 1_B\otimes \tau_B\otimes 1_B\circ \nu\otimes \nu$. In set theoretic notation with $\otimes=\times$, this would be saying $(\nu(ab)_1,\nu(ab)_2)=(\nu(a)_1\nu(b)_1,\nu(a)_2\nu(b)_2)$, writing juxtaposition for $\mu$; in other words $\nu(ab)=\nu(a)\nu(b)$ in $B\otimes B$, so that $\nu$ is a monoid homomorphism as claimed.
You can get a more thorough background on string diagram calculus from Selinger's survey here and its references.