# A Note on the Soft Group Category
## 4 Properties of the Soft Group Category  
In this section, we will show that the \(SGp\) category is a symmetric monoidal category by proving the existence of the universal properties such as the final object and a product. Theorem 7: _Let \(\{e\}\) be the trivial group, and let \(A=\{a\}\) be a singleton. Let \((F,\{a\})_{|e|}\) be a soft group such that \(F:\{a\}\to P(\{e\}),\ \ F(a)=\{e\}\). Then, \((F,\{a\})_{|e|}\) is the final object in the soft group category \(SGp\)._  
Proof: Let \((H,B)_{K}\) be an object in the soft group category \(SGp\). Then, the following diagram is commutative, where \(p(b)=a\) for every \(b\in B\) and  
\[\tikzfig{p(b)=a}\]  
states the group homomorphism defined by \(f(k)=e\) for each \(k\in K\). Thus, \((f,p)\) is the unique soft group homomorphism that can be defined from \((H,B)_{K}\) to \((F,\{a\})_{\{e\}}\). As a result, any soft group constructed by a parameter set with a single-element and a group \(\{e\}\) with one element is the final object in the soft group category \(SGp\). Proposition 2: _Let \((F_{1},A_{1})_{\text{G}_{1}}\) and \((F_{2},A_{2})_{\text{G}_{2}}\) be soft groups, and let \((U,A_{1}\times A_{2})_{\text{G}_{1}\times\text{G}_{2}}=(F_{1},A_{1})_{\text{G} _{1}}\hat{\times}(F_{2},A_{2})_{\text{G}_{2}}\) be the soft product of them. For \(i=1,2\), let \(\Pi_{i}:A_{1}\times A_{2}\to A_{i}\) be \(i\)-th projection function on the parameter sets and let \(p_{i}:\text{G}_{1}\times\text{G}_{2}\to\text{G}_{i}\) be \(i\)-th projection homomorphism on groups. Then, the map \((p_{i},\Pi_{i}):(U,A_{1}\times A_{2})_{\text{G}_{1}\times\text{G}_{2}}\to(F_{ i},G_{i})_{\text{G}_{i}}\) for each \(i=1,2\) is a soft group homomorphism._  
Proof: To prove that \((p_{i},\Pi_{i})\) is a soft group homomorphism for each \(i=1,2\), we need to show that the following diagram is commutative:  
\[\tikzfig{p(c)=a_{1}\times A_{2}\xrightarrow{U}P(G_{1}\times G_{2})}\]  
For this purpose, it is sufficient to show that the equality \(\hat{p_{i}}\circ U=F_{i}\circ\Pi_{i}\) is satisfied. For any pair \((a_{1},a_{2})\), we obtain  
\[(\hat{p_{i}}\circ U)(a_{1},a_{2}) =\hat{p_{i}}(F_{1}(a_{1})\times F_{2}(a_{2}))\] \[=F_{i}(a_{i})\] \[=F_{i}(\Pi_{i}(a_{1},a_{2}))\] \[=(F_{i}\circ\Pi_{i})(a_{1},a_{2}),\]  
and so the proof is completed. We can generalize the above proposition as follows:  
Corollary 1: _Let \((F_{1},A_{1})_{\text{G}_{1}},(F_{2},A_{2})_{\text{G}_{2}},\cdots(F_{n},A_{n})_ {\text{G}_{n}}\) be soft groups, and let \((U,A_{1}\times A_{2}\times\cdots\times A_{n})_{\text{G}_{1}\times\text{G}_{2} \cdots\times\text{G}_{n}}=(F_{1},A_{1})_{\text{G}_{1}}\hat{\times}(F_{2},A_{2 })_{\text{G}_{2}}\hat{\times}\cdots\hat{\times}(F_{n},A_{n})_{\text{G}_{n}}\) be the product of these soft groups in the sense of Definition 9. For each \(i=1,2,\cdots,n\), let \(\Pi_{i}:A_{1}\times\cdots\times A_{n}\to A_{i}\) be the \(i\)-th projection function on the parameter sets and let \(p_{i}:\text{G}_{1}\times\cdots\times\text{G}_{n}\to\text{G}_{i}\) be \(i\)-th projection homomorphism on groups. Then, the map \((p_{i},\Pi_{i}):(U,A_{1}\times\cdots\times A_{n})_{\text{G}_{1}\times\cdots \times\text{G}_{2}}\to(F_{i},A_{i})_{\text{G}_{i}}\) for each \(i=1,2,\cdots,n\) is a soft group homomorphism._  
Proof: In order to show that \((p_{i},\Pi_{i})\) is a soft group homomorphism for each \(i=1,2,\cdots,n\), we must prove that the following diagram is commutative:For any \((a_{1},a_{2},\cdots,a_{n})\in A_{1}\times A_{2}\times\cdots\times A_{n}\), we conclude that  
\[(\hat{p_{i}}\circ U)(a_{1},a_{2},\cdots,a_{n}) =\hat{p_{i}}(F_{1}(a_{1})\times F_{2}(a_{2})\times\cdots\times F_{n }(a_{n}))\] \[=F_{i}(a_{i})\] \[=F_{i}(\Pi_{i}(a_{1},a_{2},\cdots,a_{n}))\] \[=(F_{i}\circ\Pi_{i})(a_{1},a_{2},\cdots,a_{n}),\]  
which completes the proof.