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.



Name for a set of elements that fully determine a morphism
The term "base" should not be used, since, as you say, you are actually generalizing the notion of a generating set.


It is an **epi-sink**, also known as **jointly epimorphic family**. See [Joy of Cats](http://katmat.math.uni-bremen.de/acc/acc.pdf), Definition 10.62 and (dual of) Definition 10.5. A family of morphisms $(f\_i : A\_i \to A)$ is called an epi-sink when for $u,v : A \to B$ we have $\forall i (u \circ f\_i = v \circ f\_i) \implies u=v$. When the coproduct $\coprod\_i A\_i$ exists, this means that we have an epimorphism $\coprod\_{i \in I} A\_i \to A$.


If you have a terminal object $1$, morphisms $1 \to A$ are called global elements, and we can look at epi-sinks consisting of global elements of $A$.


For many categories, though, global elements are not enough. When we have a forgetful functor $U$ to $\mathbf{Set}$ with a left adjoint $F$, we have $U(-) \cong \hom(F(1),-)$, so that elements of the underlying set can be seen as morphisms on $F(1)$, and we can talk about epi-sinks on $F(1)$.


But the most general form does not put any restrictrions on the domains at all.


