For the Yoneda Embedding, where $\mathcal{T}$ is an algebraic theory,
$$
Y_\mathcal{T}: \mathcal{T}^\mathrm{op} \to Alg \mathcal{T}
$$
it is claimed that $Y_\mathcal{T}$ preserves finite coproducts. The argument provided is as follows
I am not quite sure how the last line proves that $\mathcal{T}(t_1 \times t_2, -)$ is a co-product of $\mathcal{T}(t_1, -)$ and $\mathcal{T}(t_2, -)$. It seems that all that is shown is $$ [Y_\mathcal{T}(t_1 + t_2), A] \cong [\mathcal{T}(t_1 \times t_2 , -), A]\cong A(t_1 \times t_2) \cong A(t_1) \times A(t_2) \cong [\mathcal{T}(t_1, -) , A] \times [\mathcal{T}(t_2, -) , A] \cong [\mathcal{T}(t_1, -)\times \mathcal{T}(t_2, -) , A]\cong [Y_\mathcal{T}(t_1)\times Y_\mathcal{T}( t_2), A] . $$ But I am not sure how to use that to show the claimed sentence.
Your last two isomorphims are wrong, or more precisely, the second-to-last isomorphism is wrong. The universal property of the coproduct gives you an isomorphism $$[\mathcal{T}(t_1, -) , A] \times [\mathcal{T}(t_2, -) , A] \cong [\mathcal{T}(t_1, -)+ \mathcal{T}(t_2, -) , A] = [Y_\mathcal{T}(t_1)+Y_\mathcal{T}(t_2),A],$$ which concludes the proof.