Let $C$ be the category of finitely generated algebras over a field $k$. Let $\hat{C}$ be the category of functors from $C^{op}$ to $\text{Set}$. Let $k \text{-alg}$ be the category of $k$-algebras. I would like to show
Theorem: $k \text{-alg}$ is the Ind-category of $C$.
I have a functor $F : k \text{-alg} \rightarrow \hat{C}$ sending a $k$-algebra $A$ to the functor $[-, A] : C^{op} \rightarrow \text{set}$ sending a finite $k$-algebra $B$ to the set of $k$-algebra maps $[B, A]$. This is defined on morphisms in the expected way.
I would like to show the theorem by showing that this functor is full, faithful and essentially surjective. Is that the easiest way to go about this?
To see that it is faithful, take $A, B \in \text{Obj}(k \text{-alg})$. If $f, g : A \rightarrow B$ are distinct, then they differ at some $a \in A$. Let $A'$ be the subalgebra of $A$ generated by $a$. Then $[A', f]$ and $[A', g]$ differ, so that $F(f)$ and $F(g)$ differ. This shows that $F$ is faithful.
Lemma: Each algebra in $k \text{-alg}$ is the filtered colimit in $k \text{-alg}$ of its finitely generated $k$-subalgebras.
Proof: Let $I$ be the set of finitely generated $k$-subalgebras of $A$. We make $I$ into a category whose maps are inclusions of finitely generated $k$-subalgebras of $A$. Let $F : I \rightarrow k \text{-alg}$ be the functor sending $B \in \text{Obj}(I)$ to $B \in \text{Obj}( k \text{-alg})$, and sending an inclusion of finitely generated $k$-algebras to the same inclusion in $k \text{-alg}$. Let $\eta : F \rightarrow \Delta(B)$ be a cone over $F$, where $\Delta(B) : I \rightarrow k \text{-alg}$ is the constant functor at $B$. For each $a \in A$, let $A_a$ be the algebra generated by $A$ in $a$. Define $f : A \rightarrow B$ by sending $a \in A$ to $\eta_{A_a}(a) \in B$ (we must do this).
How might I show that $F$ is essentially surjective, though?
This is not true, and your proof will not work because the subalgebra generated by $a \in A$ will not be finite-dimensional in general (e.g. consider $A = k[x]$). The correct proof will end up showing that every $k$-algebra is the filtered colimit of its finitely generated $k$-subalgebras.