2. The universal mapping property

Proposition. If $F$ is free in $V$ on $g _ 1,\dots, g _ n$ and $A$ is any algebra in $V$ and $a _ 1,\dots, a _ n \in A$, then there is a unique homomorphism $\phi:F \rightarrow A$ with $f(g _ {i}) = a _ i$ for each $i$. (In other words, you can aim the generators of $F$ at any elements of any algebra in $V$ and find a homomorphism that takes the generators there.)

Corollary 1. Up to isomorphism, there is only one free algebra in $V$ on $n$ generators.

Let us call this algebra $F _ V (n)$.

Corollary 2. Every $n$-generated algebra of $V$ is a homomorphic image of $F _ V (n)$.

Corollary 3. If $F _ V (n)$ is finite, then it is the largest $n$-generated algebra in $V$, and the only one of its size (up to isomorphism).