Determine whether the given map $\phi$ makes these two binary structures isomorphic $\langle \mathbb{Q}, \mathbb{+} \rangle$ with $\langle \mathbb{Q}, \mathbb{+} \rangle$ where $\phi(x) = x + 1$ for $x \in \mathbb{Q}$
In order to show that two binary structures are isomorphic, I need to check whether $\phi$ is one-to-one, onto, and a homorphism. 
a) one-to-one check (pass)
$$ 
\begin{align}
\phi(x) &= \phi(y) \\
x + 1 &= y + 1 \\
-1 & ~~~-1 \\
x &= y
\end{align}
$$
b) onto check (pass)
$$\phi(x+1) = (x+1) + 1 = x + 2 \textrm{ for } x \in \mathbb{Q}$$
c) homomorphism check (fail?)
$$ \phi(x + y) = (x + y) + 1 \neq \phi(x) + \phi(y) = (x+1) + (y+1) = x + y + 2$$
I think that the homomorphism check failed here and that means these two binary structures are not isomorphic for $\phi$. Did I come to the right conclusion? Am I missing any considerations in any of the checks?

Your check for a) is fine, though you might want to start with "For all $x,y\in\Bbb{Q}$...".
Your check for b) is incomplete, and has a typo (you write $\Bbb{Z}$ in stead of $\Bbb{Q}$). It would suffice to say

For all $x\in\Bbb{Q}$ we have $x=\phi(x-1)$, so $\phi$ is surjective.

Your check for c) is fine, and show that $\phi$ is indeed not an isomorphism. However, this doesn't prove that the two binary structures aren't isomorphic. In fact they are the same, so of course they are isomorphic.
In hindsight it suffices to check that c) fails, and there is no need to bother checking a) and b).
