Let $\mathbb{C}$ be some category and let $G$ be a functor $\mathcal{C}^{op} \to \mathcal{C}$. Assume that I was able to show that there is a natural isomorphism $\mathcal{C}(-, G(A \oplus B)) \Rightarrow \mathcal{C}(-, G(A) \times G(B)): \mathcal{C}^{op} \to \mathbf{Set}$, where $A$ and $B$ are two objects of $\mathcal{C}$, $\times$ is a Cartesian product in $\mathcal{C}$ and $\oplus$ a coproduct in $\mathcal{C}$.
By the Yoneda Lemma, I can deduce that $G(A \oplus B) \simeq G(A) \times G(B)$. Thus $G(A \oplus B)$ is a Cartesian product of $G(A)$ and $G(B)$ in $\mathcal{C}$. What are the projections? There are some natural candidates: $G((\iota^1)^{op})$ and $G((\iota^2)^{op})$, where $\iota^1: A \to A \oplus B$ and $\iota^2: B \to A \oplus B$ are the coprojections in $\mathcal{C}$.
How to check that it is indeed the case? Can I use the Yoneda Lemma for that?
No, your method does not work to compute the projections.
For consider $\mathcal{C} = Sets$ and $G = \Delta\mathbb{N}$ - that is, $G(A) = \mathbb{N}$, $G(f) = 1_\mathbb{N}$ for all $A, B \in \mathcal{C}$, $f : C(A, B)$.
Then there is a bijection $\tau : \mathbb{N} \to \mathbb{N} \times \mathbb{N}$ - that is, an isomorphism in the category of sets. This bijection lifts to a natural isomorphism $Sets(-, G(A \oplus B)) = Sets(-, \mathbb{N}) \to Sets(-, \mathbb{N} \times \mathbb{N}) = Sets(-, G(A) \times G(B))$.
But clearly, the maps $G(i_A) = 1_\mathbb{N} = G(i_B)$ are not the projection maps of the product.