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.