\$$\def\cat#1{{\mathcal{#1}}}\$$ \$$\def\comp#1{{\widehat{#1}}}\$$ \$$\def\conj#1{{\check{#1}}}\$$ \$$\def\id{{\mathrm{id}}}\$$ \$$\def\true{{\mathrm{true}}}\$$ \$$\def\false{{\mathrm{false}}}\$$ \$$\def\Cat#1{{\textbf{#1}}}\$$ \$$\def\BB{{\mathbb{B}}}\$$ \$$\def\ZZ{{\mathbb{Z}}}\$$ \$$\def\tn#1{{\text{#1}}}\$$
Consider the monoidal category \$$(\Cat{Set},1,\times)\$$, together with the diagram

![](https://cdn.rawgit.com/DavidRLambert/61cd3d226372556d99c1bd8631ff9833/raw/ca1d98ea6ce9b63ae1afb8c5d22d9c1d4194515a/20180814Diagram.svg)

Suppose that \$$A=B=C=D=F=G=\ZZ\$$ and \$$E=\BB=\{\true,\false\}\$$,
and suppose that \$$f_C(a)=|a|\$$, \$$f_D(a)=a*5\$$, \$$g_E(d,b)=d\leq b\$$, \$$g_F(d,b)=d-b\$$, and \$$h(c,e)=\tn{if }e\tn{ then }c\tn{ else }1-c\$$.

1. What are \$$g_E(5,3)\$$ and \$$g_F(5,3)\$$?
2. What are \$$g_E(3,5)\$$ and \$$g_F(3,5)\$$?
3. What is \$$h(5,\true)\$$?
4. What is \$$h(-5,\true)\$$?
5. What is \$$h(-5,\false)\$$?

The whole diagram now defines a function \$$A\times B\to G\times F\$$; call it \$$q\$$.

6. What are \$$q_G(-2,3)\$$ and \$$q_F(-2,3)\$$?
7. What are \$$q_G(2,3)\$$ and \$$q_F(2,3)\$$?