Show that functors preserve isomorphism. If $$a\sim a’$$ in $$\mathcal{C}$$, then $$F(a)\sim F(a’)$$ in $$\mathcal{H}$$, with $$F:\mathcal{C}\longrightarrow\mathcal{H}$$.
Solution:
Let $$f:a\longrightarrow a’$$ be a isomorphism with inverse $$f^{-1}:a’\longrightarrow a$$. By definition of functors, we have $$F(Id_{a})=Id_{F(a)}$$ and $$F(f^{-1})\circ F(f)=F(f^{-1}\circ f)$$. Thus,
\[F(f^{-1})\circ F(f)=F(f^{-1}\circ f)=F(Id_{a}).\]
The same holds for $$F(f\circ f’)=F(Id_{a’})$$.
Therefore $$F(f):F(a)\longrightarrow F(a’)$$ is an isomorphism.
