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# Exercise 15 - Chapter 3

edited June 2018

$$\large\begin{matrix} A & \overset{f}{\longrightarrow} & B \\ g \downarrow & & \downarrow h \\ C & \overset{i}{\longrightarrow} & D \end{matrix}$$ 1) Write down the ten paths in the free square category above.

2) Name two different paths that are parallel.

3) Name two different paths that are not parallel.

$$\mathrm{id}_A,\mathrm{id}_B,\mathrm{id}_C,\mathrm{id}_D,f,h,g,i,h\circ f,i\circ g$$
Among these, only two paths are parallel (i.e., have the same source and target objects): $$h\circ f$$ and $$i\circ g$$.
Any two other paths are not parallel, for example, $$f$$ and $$g$$.
Comment Source:\$$\mathrm{id}_A,\mathrm{id}_B,\mathrm{id}_C,\mathrm{id}_D,f,h,g,i,h\circ f,i\circ g\$$ Among these, only two paths are parallel (i.e., have the same source and target objects): \$$h\circ f\$$ and \$$i\circ g\$$. Any two other paths are not parallel, for example, \$$f\$$ and \$$g\$$.