Let $\mathbb{P}$ be poset.
Let $B$ be a set. We say that a $\mathbb{P}$-name $\dot{b}$ is a nice name for member of $B$ if there is a maximal antichain $A\subseteq\mathbb{P}$ and a function $h:A\rightarrow{B}$ such that $ p\Vdash\dot{b}=h(p)$ for all $p\in{A}$. Here, we abuse of the notation and say $\dot{b}=\left<{A,h}\right>$ .
Let $D, B$ be sets. We say that a $\mathbb{P}$-name $\dot{z}$ is a nice name for a function from $D$ into $B$ if there is $\left<{A_{d},h_{d}}\right>_{d \in D}$ such that $\dot{z}(d)=\left<{A_{d},h_{d}}\right>$(as a nice name) for all $d\in D$, that is, $A_{d}$ is a maximal antichain and $h_{d}:A_{d}\to B$ such that $ p\Vdash\dot{z}(d)=h_{d}(p)$ for all $p \in A_{d}$. Here, we abuse of the notation and say $\dot{z}(d)=\left<{A_{d},h_{d}}\right>_{d}$ .
Let $p \in \mathbb{P}$ and $\tau$ be a $\mathbb{P}$-name such that $p \Vdash \tau \in \check{B}$. Then there exist a nice name $\dot{b}$ for an object in $B$ such that $p \Vdash \dot{b} = \tau.$
Also if $\sigma$ be a $\mathbb{P}$-name such that $p \Vdash \sigma:\check{D}\to \check{B}$ Then there exist a nice name $\dot{z}$ for a function from $D$ into $B$ such that $p \Vdash \dot{z} = \sigma.$
I am studying the book Kunen and I'm a little confused when defining a $\mathbb{P}$-name.
Is not a question at research but can give me a suggestion how to solve this. It is a simple fact for you ,I have not been able to resolve.
A suggestion of how to define $\dot{b}$ and $\dot{z}$. Thanks