Let $\mathcal{C}$ be a category and consider $\lbrace A_i \rbrace _{i\in I}$ with $A_i\in Obj(\mathcal{C})$ and coproduct $(\amalg _{i\in I} A_i,\phi_i)$. Using the following bijection $$\text{Hom}_{\mathcal{C}}(\coprod_{i\in I} A_i,B)\cong \prod_{i \in I}\text{Hom}_\mathcal{C}(A_i,B)$$ which is functorial on $B$, and using the fact that, given an arbitrary set $I$ and a field $k$ it follows that $\dim_k(k^{I})= |k|^{|I|}$, prove the following:
If $k$ is a field and $V$ a $k$-vector of infinite dimension, then $V$ and $V^{*}=\text{Hom}_{k}(V,k)$ are not isomorphic.
My attempt:
Let $k$ be a field, then we take $\mathcal{C}$ as the category of vector spaces over $k$. Let $V$ be a $k$-vector space of infinite dimension. Let $A=\lbrace a_i \rbrace_{i\in I}$ be a $k$-basis for $V$. Then we know that $$V\cong \bigoplus_{i\in I} k$$ Now, in this category we have that the coproduct is the direct sum. So we can write $$V\cong \coprod_{i\in I} k$$ Taking a look at $$\text{Hom}_{\mathcal{C}}(\coprod_{i\in I} A_i,B)\cong \prod_{i \in I}\text{Hom}_\mathcal{C}(A_i,B)$$ We realize that if we take $A_i=k$ for all $i$ and $B=k$, then we would have $$\text{Hom}_{\mathcal{C}}(\coprod_{i\in I} k,k)\cong \prod_{i \in I}\text{Hom}_\mathcal{C}(k,k)$$ Which is now equivalent to $$\text{Hom}_{\mathcal{C}}(V,k)\cong \prod_{i \in I}\text{Hom}_\mathcal{C}(k,k)$$ And so $V^{*} \cong \prod_{i \in I}\text{Hom}_\mathcal{C}(k,k)$. Now, we know that for a given vector space $V$ over $k$ with $k$-basis $\lbrace v_i \rbrace_{i\in I}$, we have that $\text{End}(V) \cong \bigoplus_{i\in I} V$. Since $\text{Hom}_\mathcal{C}(k,k) = \text{End}(k)$ and $k$ is just $k$ as a vector space over $k$, then $\text{Hom}_\mathcal{C}(k,k) \cong k$, and so, $\cong \prod_{i \in I}\text{Hom}_\mathcal{C}(k,k) \cong k^I$, which gives us $V^{*} \cong K^I$.
Now, we have $|V^{*}|=|k|^{|I|}$, and we know that for an vector space over $k$, we have $|V|=|k|·\text{dim}(V)$ and $\text{dim}(V) = |A|=|I|$, and so for any $k$-vector space different than $k$, we would have $$|V|=|k|·|I|$$ $$|V^{*}|=|k|^{|I|}$$ from which it follows that $V$ and $V^{*}$ could never be isomorphic. Is this a correct reasoning?
I would also want to understand what is the role of functorial on B in this context. Also, I'm not sure if I'm missing something yet to check, is there anything else that I should see for the reasoning to be complete?
Thanks in advice for the replies.