Looks good, Keith... but I'm an old fuddy-duddy: I like to see everything spelled out in detail. Preservation of identities says that

$\mathrm{hom}(1_{c,c'}) = 1_{\mathrm{hom}(c,c')}$

so I'd want to see an argument leading up to this conclusion, and preservation of composition says that

$\mathrm{hom}((f,g) \circ (l,j)) = \mathrm{hom}(f,g) \circ \mathrm{hom}(l,j)$

so I'd want to see an argument leading up to this. You've got a lot of the building-blocks there!