QUESTION: I would like some hints for part b.I am unsure of how to show that $\widetilde{T}$ is surjective or injective for that matter. My construction for $\widetilde{T}$ is $\widetilde{T}((v_i)_{i\in I})=\sum_{i\in I} j'(v_i)$. I feel that there may be some problems with my construction. of $\widetilde{T}$.
Put $Q=\bigoplus_{i^\prime\in I} V_{i^\prime}$. Then there is a unique $\tilde T:Q\to Q^\prime$ such that $\tilde T\circ j_i=j_i^\prime$ for every $i\in I$. There is also a unique $T^\prime:Q^\prime\to Q$ such that $T^\prime\circ j_i^\prime=j_i$ for all $i\in I$. But now $\tilde T\circ T^\prime$ is the unique morphism satisfying $\tilde T\circ T^\prime\circ j_i^\prime=\tilde T\circ j_i=j_i^\prime$ so $\tilde T\circ T^\prime=\DeclareMathOperator{id}{id}\id_{Q^\prime}$. Similarly $T^\prime\circ \tilde T=\id_Q$. Hence $\tilde T:Q\to Q^\prime$ is an isomorphism.