TITLE: Pushout from initial object isomorphic to coproduct
QUESTION [5 upvotes]: Let $C$ be a category with an initial object $0$, pushouts, and coproducts.
If $G$ and $H$ are two objects, I want to know whether $G\sqcup H$ and $G\sqcup_0 H$ are
isomorphic.
Letting $i_1$ and $i_2$ be the morphisms from $G$ and $H$ into $G\sqcup H$, and letting
$\phi_1$ and $\phi_2$ be the morphisms from $G$ and $H$ into $G\sqcup_0 H$, I know there 
are two unique morphisms $i:G\sqcup_0 H\rightarrow G\sqcup H$ and $\phi:G\sqcup H\rightarrow
G\sqcup_0 H$ such that $\phi_1=\phi\circ i_1$, $\phi_2=\phi\circ i_2$, $i_1=i\circ \phi_1$,
and $i_2=i\circ \phi_2$.
That is, the following diagram is commutative:
$\hskip1in$ 
I believe I need to show $\phi\circ i=id_{G\sqcup H}$ and that $i\circ \phi=id_{G\sqcup_0 H}$,
as they are the only morphisms showing up. But now I am stuck.
This is just for personal interest. I'd appreciate any help finishing the proof, or 
demonstrating a counter-example.

REPLY [6 votes]: As almost always the Yoneda Lemma offers a one-line proof.
$\small \hom(G \sqcup H,K) = \hom(G,K) \times \hom(H,K) = \hom(G,K) \times_{\hom(0,K)} \hom(H,K) = \hom(G \sqcup_0 H,K)$