Define the direct sum of two vector spaces as per Wikipedia:
https://en.wikipedia.org/wiki/Direct_sum_of_modules#Construction_for_two_vector_spaces
Why does this satisfy the definition of the categorical coproduct?
https://en.wikipedia.org/wiki/Coproduct
My best guess is that the “canonical injections” can be taken to be the the embeddings $j_i:V_i\to \bigoplus_k V_k$ since they are morphisms (because they are linear maps) and they yield that whenever $h_i:V_i \to Z$ are linear maps into a vector space $Z$, there exists a unique linear map $H: \bigoplus_k V_k\to Z$ such that $$H\circ j_i = h_i, \forall i.\quad (1)$$ Indeed, defining $H(v_1\oplus v_2 \oplus …):= h_1(v_1)+h_2(v_2)+…$ we get existence of $H$ and for uniqueness we realise that, supposing $\tilde{H}$ another linear map such that (1) holds: $\tilde{H}(v_1\oplus v_2 \oplus …)=\tilde{H}(v_1\oplus 0 \oplus …) + \tilde{H}(0\oplus v_2 \oplus0 \oplus …)$$= \sum_k \tilde{H}(…\oplus 0 \oplus v_k\oplus 0 \oplus …)= \sum_k \tilde{H}\circ j_k(v_k)= \sum_k h_k (v_k)=H(v_1\oplus v_2 \oplus …)$.
Is my proof accurate?