In a concrete category (i.e., where the morphisms are functions between sets), I define a **base** of an object $A$ to be a set of elements $M$ of $A$ such that for any morphisms $F,G:A\to B$ that coincide on $M$, we have $F=G$.

**Question:** Is there an established name for a **base** in that sense?

**Examples:** In the category of vectors spaces, generating sets are bases. In the category of sets, $A$ is the only base of $A$.

**Note:** The above definition does not really need a concrete category (an initial object is enough), but I decided to formulate it in a concrete category for simplicity.

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