Cheuk Man Hwang wrote:

> 1) **Preservation of composition:**

> Suppose \\(h\in\mathcal{C}(c, c')\\) and \\((f,g)\\) is a morphism from \\((c, c')\\) to \\((d, d')\\) and \\((l, j)\\) is a morphism from \\((d, d')\\) to \\((e, e')\\). Then we have

> \[
\begin{array}{ccc}
\mathrm{hom}\big((l, j)\circ(f, g)\big) h &=&\mathrm{hom}\big((l\circ_{op} f, j\circ g)\big)h\\\\
&:=&(j\circ g)\circ h \circ (l\circ_{op} f)\\\\
&=&(j\circ g)\circ h \circ (f\circ l)\\\\
&=&j\circ (g\circ h \circ f)\circ l\\\\
&:=&\mathrm{hom}\big((l, j)\big) (g\circ h \circ f)\\\\
&:=&\mathrm{hom}\big((l, j)\big) \circ \mathrm{hom}\big((f, g)\big) h\\\\
\end{array}
\]

> This shows that \\(\mathrm{hom}\big((l, j)\circ(f, g)\big)=\mathrm{hom}\big((l, j)\big) \circ \mathrm{hom}\big((f, g)\big)\\).

Great! There's something nice about not skipping any steps and seeing how all the rules get used.