## What is “Free”

**Update**.Suppose $\mathbf{B}$ is a category, $\mathbf{A}$ is a subcategory of $\mathbf{B}$, and $i:X\to G$ is an injective morphism. If for each object $Y$ of $\mathbf{A}$ and each morphism $f:X\to Y$ there is a unique morphism $\tilde{f}:G\to Y$ such that the following diagram is commutative:
\begin{xy}
\xymatrix{
X\ar[r]^f \ar[d]_i&Y\\
G\ar[ur]_{\tilde{f}} & ,
}
\end{xy}
then $G$ is said to be a *free* object with a *basis* $X$.
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Suppose $\mathbf{A}$ is a category, $G$ is an object of $\mathbf{A}$. If there is a parent category $\mathbf{B}$ of $\mathbf{A}$ with an object $X$ and an injective morphism $i:X\to G$ satisfying that for any object $Y$ of $\mathbf{A}$ and each morphism $f:X\to Y$ there is a unique morphism $\tilde{f}:G\to Y$ such that the following diagram is commutative:
\begin{xy}
\xymatrix{
X\ar[r]^f \ar[d]_i&Y\\
G\ar[ur]_{\tilde{f}} & ,
}
\end{xy}
then $G$ is said to be a *free* object.
(discussed with S.Q.Huang)