In some categories, to verify that a map $f : X \to Y$ is a morphism, it suffices to check only a generating set for $Y$ (or, rather, a generating set for some structure on $Y$ such as a topology or a $\sigma$-algebra).
Here are two examples:
If $(X, \mathcal{T})$ and $(Y, \mathcal{T}')$ are topological spaces, in order to show that a function $f : X \to Y$ is continuous, it suffices to check that for every element $S$ of a basis or subbasis of $\mathcal{T}'$, we have $f^{-1}(S) \in \mathcal{T}$.
If $(X, \mathcal{A})$ and $(Y, \mathcal{B})$ are measurable spaces, in order to show that $f: X \to Y$ is measurable, it suffices to check that for every element $B$ of a generating set for $\mathcal{B}$, we have $f^{-1}(B) \in \mathcal{A}$.
Is there any deeper meaning to this phenomenon, or does it simply arise "by nature"? More concretely, are there any nontrivial properties that a category must have in order for it to be true that in order to verify that a map is a morphism, it suffices to check a generating set for the codomain of the map?
As Zhen Lin noted, a category with this property must be concrete, but what else can be said about it?