In the paper "Homotopy, homology, and persistent homology using closure spaces Peter Bubenik, Nikola Milićević" HypGph denote the category of hypergraphs and hypergraph homomorphisms. In the paper Decorated Cospans in section 2.2 (alos in may other papers) they defined category of hypergraphs as follows: 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}
How are they same? Can someone explain? here $\mu: X \to X \otimes X, \delta:X\otimes X \to X, \eta:I \to X, \epsilon: X \to I,$ and $\sigma$ is braiding.
The equations for $\mu_{X\otimes Y},\eta_{X\otimes Y}$ etc. that you write are rather the definition of a (commutative) Frobenius structure on $X\otimes Y$ based on the (commutative) Frobenius structures on $X$ and $Y$ respectively.
In other words, the tensor product of two (commutative) Frobenius algebras in a symmetric monoidal category inherits a (commutative) Frobenius algebra structure. (Check that they indeed satisfy the Frobenius axioms).