In the paper Decorated Cospsans by Brendan Fong, they gave the definition
Let (X) be an object in a symmetric monoidal category with a special commutative Frobenius structure $(X, \mu_X, \delta_X, \eta_X, \epsilon_X)$. The following equations hold:
\begin{align} \mu_{X \otimes Y} &= (\mu_X \otimes \mu_Y) \circ (1_X \otimes \sigma_{YX} \otimes 1_Y) \\ \eta_{X \otimes Y} &= \eta_X \otimes \eta_Y \\ \delta_{X \otimes Y} &= (1_X \otimes \sigma_{XY} \otimes 1_Y) \circ (\delta_X \otimes \delta_Y) \\ \epsilon_{X \otimes Y} &= \epsilon_X \otimes \epsilon_Y \end{align}
and in the paper Hypergraph Categories by Brendan Fong and David I. Spivak they defined by string diagrams in the following way in image. But isn't the first diagram is of the only part $1_X \otimes \sigma_{YX} \otimes 1_Y$? where did the $\mu_X \otimes \mu_Y$ part go?
i have also added the string diagram which I drew and acoording to me should be made.
Look at the right-hand side of the first diagram,
The crossing of two wires is $\sigma_{YX}$. So that right-hand diagram shows $1_X\otimes \sigma_{YX}\otimes 1_Y$
composed with $\mu_X\otimes \mu_Y$
So the overall diagram is $(\mu_X\otimes \mu_Y)\circ(1_X\otimes \sigma_{YX}\otimes 1_Y)$.
Note that the $\circ$ notation swaps the order of the two morphisms. Some people write composition the other way around, so that this expression would be written $(1_X\otimes \sigma_{YX}\otimes 1_Y);(\mu_X\otimes \mu_Y)$. That way the order of the algebraic notation matches that of the diagram.