A rough definition of a symmetric monoidal structure on a category \(\mathcal{C}\)
Two additional constituents
An object \(I \in Ob(\mathcal{C})\) called the monoidal unit
A functor \(\mathcal{C}\times \mathcal{C}\xrightarrow{\otimes}\mathcal{C}\) called the monoidal product
Subject to the well-behaved, natural isomorphisms
\(I \otimes c \overset{\lambda_c}\cong c\)
\(c \otimes I \overset{\rho_c}\cong c\)
\((c \otimes d)\otimes e \overset{\alpha_{c,d,e}}\cong c \otimes (d\otimes e)\)
\(c \otimes d \overset{\sigma_{c,d}}\cong d \otimes c\)
A category equipped with these is a symmetric monoidal category