Consider the monoidal category \((\mathbf{Set},1,\times)\) together with the following diagram TODO - NEED TO COPY HERE
\(A=B=C=D=F=G=\mathbb{Z}\) and \(E=\mathbb{B}\)
\(f_C(a)=|a|\),
\(f_D(a)=a*5\),
\(g_E(d,b)=d\leq b\)
\(g_F(d,b)=d-b\)
\(h(c,e)=\text{if }e\text{ then }c\text{ else }1-c\)
Suppose the whole diagram defines a function \(A \times B \xrightarrow{q} G \times F\)
Answer:
What are \(g_E(5,3)\) and \(g_F(5,3)\)?
What are \(g_E(3,5)\) and \(g_F(3,5)\)?
What is \(h(5,true)\)?
What is \(h(-5,true)\)?
What is \(h(-5,false)\)?
What are \(q_G(-2,3)\) and \(q_F(-2,3)\)?
What are \(q_G(2,3)\) and \(q_F(2,3)\)?
\(False,\ 2\)
\(True,\ -2\)
\(5\)
\(-5\)
\(6\)
\((2,-13)\) ... \(a\mapsto -2,\ b \mapsto 3,\ c\mapsto 2,\ d\mapsto -10,\ e\mapsto true,\ f\mapsto -13, g \mapsto 2\)
\((-1,7)\) ... \(a\mapsto 2,\ b \mapsto 3,\ c \mapsto 2,\ d\mapsto 10,\ e\mapsto false, f\mapsto 7, g\mapsto -1\)