Equality of objects in a concrete category

Question

A concrete category is a pair $(C,U)$ such that $U:C\to \mathbf{Set}$ is a faithful functor. It means that for any $Y,Z\in \mathrm{ob}(C)$, and a function between the sets $f:U(Y)\to U(Z)$ - if exists a morphism $\phi :Y\to Z$ with $U(\phi)=f$, then it's unique.

Namely, given a function $U(Y)\to U(Z)$, we can say that the function is either a morphism $Y\to Z$ or it is not.

I've noticed the following property in a lot of concrete categories:

$(*)$ if $Y,Z\in \mathrm{ob}(C)$, and $U(Y)=U(Z)$ (namely exist the identity functions $U(Y)\to U(Z),U(Z)\to U(Y)$) so, if both identity functions are morphisms, then $Y=Z$.

A. This is my actual question: is there a name for this property?

B. This is the motivation for me asking this: I'm using these definitions for subobject and quotient object (which are similar to an answer I got in a question but not exactly the same, so they might be wrong):

Let $X,Y\in \mathrm{ob}(C)$. $Y$ is a subobject of $X$ if

1. $U(Y)\subseteq U(X)$, so we will denote $\iota: U(Y)\to U(X)$ the embedding function;
2. For any $A\in \mathrm{ob}(C)$ and a function $f:U(A)\to U(Y)$, $f$ is a morphism $A\to Y$ iff $\iota\circ f$ is a morphism $A\to X$.

$Y$ is a quotient object of $X$ if

1. $U(Y)$ is a quotient of $U(X)$ by an equivalence relation, so we will denote $q: U(X)\to U(Y)$ the quotient map;
2. For any $A\in \mathrm{ob}(C)$ and a function $f:U(Y)\to U(A)$, $f$ is a morphism $Y\to A$ iff $f\circ q$ is a morphism $X\to A$.

Now!!!! The moment we've been waiting for!!1 Just like a morphism between two objects in a concrete category is uniquely determined by the underlying function ($U$ is faithful), I want to say that a subobject is uniquely determined by the underlying set. (Same for quotient object - uniquely determined by the underlying set namely by the equivalence relation).

Then this is where I got:

If $Y$ is in fact a subobject of $X$, then the embedding function itself is a morphism: because the identity function $id_{U(Y)}:U(Y)\to U(Y)$ is certainty a morphism $Y\to Y$, therefore $\iota\circ id_{U(Y)}=\iota$ is a morphism too.

Assume $Y,Z\in \mathrm{ob}(C)$ and $U(Y)=U(Z)$.

If $Y$ is a subobject of $X$, then exists the identity function $f:U(Z)\to U(Y)$, and $\iota_{Z}=\iota_{Y}\circ f$ therefore $\iota_{Y}\circ f$ is a morphism. Thus, the identity function $U(Z)\to U(Y)$ is a morphism.

If $Z$ is a subobject of $X$, then symmetrically the identity function $U(Y)\to U(Z)$ is a morphism.

This is where I want to claim that $Y=Z$, so I need property $(*)$.