**Puzzle 161**

![homfunctor preservation rules](http://aether.co.kr/images/homfunctor_preservation_example.svg)

I apologize for taking the liberty to rename objects and morphisms as shown in the diagram above for it was easier to work with for me. I have also taken out all diagonal morphisms and identity morphism minus the one shown for simplicity of proving the preservation rules.

*Unit Preservation*:

So we start with identities \$$1_a:a \rightarrow a\$$ and \$$1_b:b \rightarrow b\$$ and hope that when we take the homfunctor \$$C(1_a , 1_b)\$$, it is the identity for objects in homfunctor, \$$1_{C(a,b)}\$$. First take the product \$$(1_a, 1_b) = (a \rightarrow a, b \rightarrow b)\$$ and then by taking the homfunctor, we get the morphism \$$C(1_a , 1_b) : C(a,b) \rightarrow C(a,b)=1_{C(a,b)}\$$ which is too trivial to see the details.

*Composition Preservation*:

We need to show \$$C(f,i) \circ C(g,h)= C(g \circ f, i \circ h)\$$. The left hand side is the composition shown in the diagram on the right which takes the object \$$C(a,b) \rightarrow C(a',b) \rightarrow C(a'',b'')\$$. On the right side, we get \$$C(g \circ f, i \circ h) = C(a'' \rightarrow a' \rightarrow a, b \rightarrow b' \rightarrow b'') = C(a'' \rightarrow a, b \rightarrow b'')\$$ which is just the morphism \$$C(a,b) \rightarrow C(a'',b'')\$$ as you can see by the composition \$$i \circ h \circ C(a,b) \circ g \circ f:a'' \rightarrow b''\$$.

For the newbies like I, while doing this puzzle found this to be helpful when translating from diagrams to equations.

![homfunctor equation](http://aether.co.kr/images/homfunctor_equation.svg)