Sorry for breaking the harmony of MO with a easy and silly questions. I have stuck on elementary category-theoretic reasoning about subcoalgebras, namely:

Let $F:\mathcal{Set}\to \mathcal{Set}$ be a functor. An $F$-coalgebra is a pair $\mathcal{A}=(A,\alpha)$ where $\alpha:A\to F(A)$ is an arbitrary map. Given $F$-coalgebras $\mathcal{A}=(A,\alpha)$ and $\mathcal{B}=(B,\beta)$, a homomorphism $\varphi:\mathcal{A}\to \mathcal{B}$ is a map $\varphi:A\to B$ which make the obvious corresponding square commute.

A subset $U\subseteq A$ is called closed, if an $F$-coalgebra structure $\mathcal{U}=(U,\delta)$ can be defined on $U$ so that the natural inclusion $\subseteq :U\to A$ is a homomorphism. In this case $\mathcal{U}$ is called subcoalgebra of $\mathcal{A}$.

**A coalgebra literature indicates that coalgebra structure map $\delta$ is easily seen to be unique.**

I have stuck on uniqueness, what is the reason, how it can be seen? Is it category-theoretic reason or rather set-theoretic?