Is there a name for the following notion in category theory, which weakens fullness?
A subcategory $\mathcal A$ of $\mathcal B$ has the property that for all objects $x,y \in \mathcal A$, if there is an arrow $f : x \to y$ in $\mathcal B$ then there is one in $\mathcal A$ also.
Bump: I have chosen the name honest subcategory. Why is this an interesting notion? Consider a small category $\mathcal C$ which is in some level of the set-theoretic hierarchy $V_\alpha$. Let $M \prec V_\alpha$ be an elementary submodel with $\mathcal C \in M$. Then $\mathcal C \cap M$ is an honest subcategory of $\mathcal C$, but not necessarily full. The point is that honesty is witnessed by first-order properties, while fullness is not.